Mathematics and Its Applications
Limit Theory for Mixing Dependent Random
Variables
LIN ZHENGYAN
and
LU CHUANRONG
Department of Mathematics,
Hangzhou University,
Hangzhou, People's Republic of China
For many practical problems, observations are not independent. In this book,
limit behaviour of an important kind of dependent random variables, the so-called
mixing random variables, is studied. Many profound results are given, which
cover recent developments in this subject, such as basic properties of mixing
variables, powerful probability and moment inequalities, weak convergence and
strong convergence (approximation), limit behaviour of some statistics with a
mixing sample, and many useful tools are provided.
Audience
This volume will be of interest to researchers and graduate students in the field
of probability and statistics, whose work involves dependent data (variables).
ISBN 0-7923-4219-4
Science Press
KLUWER ACADEMIC PUBLISHERS ΜΑΙΑ378
780792ll342199l

Mathematics and Its Application
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science,
Amsterdam, The Netherlands
Volume 378

Limit Theory for
Mixing Dependent Random Variables
by
Lin Zhengyan
and
Lu Chuanrong
Department of Mathematics,
Hangzhou University,
Hangzhou, The People's Republic of China
й
Science Press
New York/Beijing
Kluwer Academic Publishers
Dordrecht/Boston/London

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ISBN 1-880132-27-5
ISBN 0-7923-4219-4
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SERIES EDITOR'S PREFACE
'Et moi, ..., si j'avait su comment en revenir, One service methematics has rendered the
je n'y serais point alle.' human race. It has put common sense back
Jules Verne where it belongs, on the topmost shelf next
to the dusty canister labelled 'discarded non-
The series is divergent; therefore we may be ,
secse .
able to do something with it. τ-· · m r> »
Eric T. Bell
O.Heaviside
Mathematics is a tool for thought. A highly necessary tool in a world where both
feedback and nonlinearities abound. Similarly, all kinds of parts of mathematics serve
as tools for other parts and for other sciences.
Applying a simple rewriting rule to the quote on the right above one finds such
statements as: 'One service topology has rendered mathematical physics ...'; 'One service
logic has rendered computer science ...'; 'One service category theory has rendered
mathematics ...'. All arguable true. And all statements obtainable this way form part
of the raison d'etre of this series.
This series, Mathematics and Its Applications, started in 1977. Now that over one
hundred volumes have appeared it seems opportune to reexamine its scope. At the
time I wrote
"Growing specialization and diversification have brought a host of monographs and
textbooks on increasingly specialized topics. However, the 'tree' of knowledge of
mathematics and related fields does not grow only by putting forth new branches. It also
happens, quite often in fact, that branches which were thought to be completely
disparate are suddenly seen to be related. Further, the kind and level of sophistication of
mathematics applied in various sciences has changed drastically in recent years: measure
theory is used (non-trivially) in regional and theoretical economics; algebraic geometry
interacts with physics; the Minkowsky lemma, coding theory and the structure of water
meet one another in packing and covering theory; quantume fields, crystal defects anf
mathematical programming profit from homotopy theory; Lie algebras are relevant to
filtering; and prediction and electrial engineering can use Stein spaces. And in
addition to this there are such new emerging subdisciplines as "experimental methematics',
'CFD', 'completely integrable systems', 'chaos, synergetics and large-scale order', which
are almost impossible to fit into the existing classification schemes. They draw upon
widely different sections of mathematics."
By and large, all this this still applies today. It is still true that at first sight
mathematics seems rather fragmented and that to find, see, and exploit the deeper underlying
interrelations more effort is needed and so are books that can help mathematicians and
scientists do so. Accordingly MIA will continue to try to make such book available.
If anything, the description I gave in 1977 is now an understatement. To the examples
of interaction areas one should add string theory where Riemann surfaces, algebraic
geometry, modular functions, knots, quantum field theory, Kac-Moody algebras,
monstrous moonshine (and more) all come together. And to the examples of things which
can be usefully applied let me add the topic 'finite geometry'; a combination of words
which sounds like it might not even exist, let alone be applicable. And yet it is being
applied: to statistics via designs, to radar/sonar detection arrays (via finite projective
planes), and to bus connections of VLSI chips (via difference sets). There seems to

be no part of (so-called pure) mathematics that is not in immediate danger of being
applied. And, accordingly, the applied mathematician needs to be aware of much more.
Besides analysis and numerics, the traditional workhorses, he may need all kinds of
combinatorics, algebra, probability, and so on.
In addition, the applied scientist needs to cope increasingly with the nonlinear world
and the extra mathematical sophistication that this requires. For that is where the
rewards are. Linear models are honest and a bit sad and depressing: proportional
efforts and results. It is in the nonlinear world that infinitesimal inputs may result in
macroscopic outputs (or vice versa). To appreciate what I sm hinting at; if electronics
were linear we would have no fun with transistors and computers; we would have no
TV; in fact you would not be reading these lines.
There is also no safety in ignoring such outlandish things as nonstandard analysis,
superspace and anticommuting integration, p-adic and ultrametric space. All three have
applications in both electrical engineering and physics. Once, complex numbers were
equally outlandish, but they frequently proved the shortest path between 'real' results.
Similarly, the first two topics named have already provided a number of 'wormhole'
paths. There is no telling where all this is leading-fortunately.
Thus the original scope of the series, which for various (sound) reasons now comprises
five subseries: white (Japan), yellow (China), red (USSR), blue (Eastern Europe), and
green (everything else), still applies. It has been enlarged a bit to include book treating
of the tools from one subdiscipline which are used in others. Thus the series still aims
at books dealing with:
- a central concept which plays an improtant role in several different mathematical and/or
scientific specialization areas;
- New applications of the results and ideas from one area of scientific endeavour into
another;
- influences which the results, problems and concepts of one field of enquiry have, and
have had, on the development of another.
The present volume, one of the first in the 'Chinese subseries' of MIA, also
appropriately enough, one dealing with fundamental issues: interrelations between logic and
computer science. The advent of computers has sparked off revived interest in a host
of fundamental issues in science and mathematics such as computability, recursiveness,
computational complexity and automated theorem proving to which latter topic ths
author has made seminal contributions for which he was awarded the ATP prize in
1982.
It is a pleasure to welcome this volume in this series.
The shortest path between two truths in the
real domain passes through the complex
domain
J. Hadamard
La physique ne nous donne pas seulement
;Occasion de resoudr des problemes ... elle
nous fait presentir la solution.
H. Poincare
Never lend books, for no one ever returns
them; the only books I have in my library
are books that other folk have lent me.
Anatole France
The function of an expert is not to be more
right than other people, but to be wrong for
more sophisticated reasons.
David Butler
Bussum, August 1989
Michiel Hazewinkel

Preface
The classical limit theorems of probability theory for independent
random variables had been developed successfully in the thirties and forties.
The basic results were summed up in Gnedenko and Kolmogorov's
monograph "Limit Distributions for Sums of Indenpendent Random
Variables " (1954) and Petrov's monograph "Sums of Independent
Random Variables " (1975) . The modern limit theorems of probability theory,
such as weak convergence of probability measures and strong
approximations etc, have been studied by many authors since the fifties. The limit
theory for weakly dependent random variables was also discussed deeply.
In fact, the limit distributions of,sums for non-independent random
variables were studied early by some probabilitists and statisticians, such as
Bernstein (1927), Hopf (1937), Hoeffding and Robbins (1948), etc. The
dependence of random variables as a concept is developed not only in some
branches of probability theory and mathematical statistics, such as Markov
chains, random field theory and time series analysis, etc, but also appears
in many practical problems. Although the assumption of independence is
reasonable sometimes, it is difficult to check the independence of a sample.
Moreover in many practical problems, the samples are not independent
observations.
The definition of strong mixing (α-mixing) was first introduced by
Rosenblatt (1956). Ibragimov (1959), Rozanov and Volconski (1959) also
introduced this concept independently at the same time as they introduced
the definition of ^-mixing. The definition of />mixing was introduced by
Kolmogorov and Rozanov (1960). All these concepts describe the
asymptotic independence of random variables when the difference of their indices
goes to infinity. The 1971 monograph by Ibragimov and Linnik,
"Independent and Stationary Sequence of Random Variables " summed up
main results of convergence in distribution for a mixing sequence up to the
sixties. Since the theory of weak convergence of probability measures
appears, particularly, following the monograph by Billingsley," Convergence
of Probability Measures "(1968), the weak convergence for a sequence of
mixing random variables attracts the attentions of many authors and some

viii
Preface
ideal results have been obtained. The theory of strong approximations for
a sequence of dependent random variables is discussed systematically in
Philipp and Stout's monograph"AImost Sure Invariance Principle for
Partial Sums of Weakly Dependent Random Variables " (1975).
The results of this monograph have been improved comprehensively by us.
The modern limit theory for a sequence of mixing random variables has
been studied deeply by many authors. This book will introduce them
comprehensively, including Z. Y. Lin, C. R. Lu and Q. M. Shao's work for
the weak convergence and strong approximations.
The book consists of four parts. The first part contains two chapters.
We shall introduce the definitions of various mixing sequences and give a
series of inequalities for mixing random variables, some of them are due to
Shao. These inequalities are indispensable tools for the proofs of various
limit theorems.
In the second part, which is separated into five chapters, the weak
convergence, Berry-Esseen inequality and the rate of weak convergence are
discussed. Some ideal results, such as weak convergence of /9-mixing
sequences, will be introduced.
In the third part, the almost sure convergence and strong approximations
for the mixing random variables are studied. There are four chapters in this
part. Some best results will be presented, such as strong approximations
of partial sums for mixing sequences, which are done by Shao and Lu; the
limiting behaviour of the increments of partial sums for a mixing sequence
is obtained by Lin, et al.
In the fourth part, the weak convergence and strong approximations for
some statistics with mixing dependent samples and some other kinds of
dependent random variables are studied. Most results are profound.
Our best thanks are due to Dr. Q. M. Shao, whose results enrich greatly
the book. We also want to thank all colleagues who help us to complete
the book. We express our most gratitude to National Science Foundation
of China and Zhejiang Province for their financial supports as well.
Lin, Ζ. Υ.
Lu, С R.
Hangzhou University, May 1996

Contents
Preface vii
Part I Introduction 1
Chapter 1 Definitions and Basic Inequalities 3
1.1 Definitions 3
1.2 Basic inequalities 7
Chapter 2 Moment Estimations of Partial Sums 15
2.1 Variances of partial sums 15
2.2 Further inequalitis 24
Part II Weak Convergence 39
Chapter 3 Weak Convergence for α-mixing Sequences 41
3.1 Necessary and sufficient conditions for the CLT 41
3.2 Sufficient conditions for CLT and WIP 51
3.3 The CLT and WIP when the variance is infinite 63
Chapter 4 Weak Convergence for ρ -mixing Sequences 71
4.1 The WIP when the moments of order 2 are finite 73
4.2 The WIP when moments of higher than two orders 78
4.3 A generalized result when moments of
higher than two orders 88
4.4 The WIP when the variance is infinite 104
Chapter 5 Weak Convergence for φ -mixing Sequences 123
5.1 The WIP when the moments of order 2 are finite 124
5.2 The Ibragimov-Linnik-Iosifescu conjecture 136
Chapter 6 Weak Convergence for Mixing Random Fields 141
6.1 The CLT for mixing random fields 141
6.2 Convergence of finite dimensional distributions 149
6.3 Tightness 160

X Contents
Chapter 7 The Berry-Esseen Inequality and the Rate
of Weak Convergence 169
7.1 Rate of convergence in distribution for a-mixing
and /o-mixing sequences 169
7.2 The rate of weak convergence for a (^-mixing sequence 181
Part III Almost Sure Convergence and
Strong Approximations 189
Chapter 8 Laws of Large Numbers and Complete Convergence 191
8.1 Weak law of large numbers 191
8.2 Strong laws of large numbers 199
8.3 Complete convergence for (^-mixing sequences 201
8.4 Complete convergence for /o-mixing sequences 208
8.5 Complete convergence for α-mixing sequences 222
8.6 A further discussion on the complete convergence 231
Chapter 9 Strong Approximations 241
9.1 Strong approximations for a <£-mixing sequence 241
9.2 Strong approximations for a p-mixing sequence 247
9.3 Strong approximations for a α-mixing sequence 263
Chapter 10 The Increments of Partial Sums 269
10.1 Some lemmas 269
10.2 How big are the increments when the moment
generation functions exist? 277
10.3 How big are the increments when the moment
generating functions do not exist? 283
Chapter 11 Strong Approximations· for Mixing Random Fields 287
11.1 Strong approximations of a <£>-mixing random field 288
11.2 Strong approximations of a a -mixing random fields 301
Part IV Statistics of a Dependent Sample 309
Chapter 12 Empirical Processes : 311
12.1 Weak convergence 312
12.2 Weighted weak convergence 317
12.3 Strong approximations 331
12.4 Moduli of continuity of empirical processes 341

Contents XI
Chapter 13 Convergence of Some Statistics with
a Mixing Sample 347
13.1 U-Statistics 347
13.2 Error variance estimations in linear models 360
13.3 Density estimations 369
Chapter 14 Strong Approximations for Other Kinds of
Dependent Random Variables 379
14.1 Lacunary trigonometric series with weights 379
14.2 A class of Gaussian sequences 392
14.3 The non-negative additive functional of a Markov process ... 397
Appendix 406
References 409
Index :: 425

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Part I Introduction
In this part, we introduce some common and important definitions of
weakly dependent random variables, establish some bounds of covariances
for the various mixing sequences, and also discuss the relations between
each other for different definitions. These will be given in Chapter 1.
In Chapter 2, we give the estimations of some kind of moments of
partial sums of a mixing sequence, which play important roles in the limit
theorems and will be used often.

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Chapter 1 Definitions and Basic Inequalities
In this book, we always assume that {Xn, η > 1} is a sequence of
random variables defined on a probability space (Ω,.77, Ρ). There are many
ways to describe weak dependence or asymptotic independence of {Xn}·
In Section 1.1, we give some common and important definitions of this
kind. In Section 1.2, some basic inequalities on covariances of {Xn} are
established, which are useful for studying limit properties of {Xn}· In
these sections, we also discuss the relations between each other for different
definitions.
1.1 Definitions
Let A and В be sub-a-fields of T, LP(A) a set of all A-measurable
random variables with p-th moments. Define
a(A,B) = sup \P(AB) - P(A)P(B)l
Аел,вев
\EXY - EXEY\
p(A,B)= sup J— =-+,
xeL2(A),YeL2(B) ν VarX УагУ
φ(Α,Β)= sup \P(B\A) - P(B)l
АеА,вев,Р(А)>о
«*»>= -p |P(AB>-;((T(B)'
AeA,BeB,P(A)P(B)>0 ^ΚΑΗ\ΰ)
β(Α,Β) = E(tv<iTBeB\P(B\A) - P(S)|),
ил*\ \ΕΧΥ-ΕΧΕΥ\
\(А9В) = sup l-— rr— ί,
XeL1/a(A),YeL1/fi(B) \\Λ \\ΐ/α\\Υ\\1/β
where tvar means total variation and ||-X"||P = (i£|X|p) . Let T\ =
σ(Χ%ι α < г < Ь), Ζ a set of all integers, Z+ a set of all non-negative
integers, N a set of all positive integers. Some commom and important

4
Chapter 1 Definitions and Basic Inequalities
definitions of mixing sequences are as follows:
Definition 1.1.1. A sequence {Xn, η > 1} is said to be a-mixing or
strong mixing if
a(n) = supa(^,^+n) —► 0 asn->oo.
fcGN
Definition 1.1.2. A sequence {Xn, η > 1} is said to be p-mixing if
p(n) = supp(JFf ,jF°^n) -+0 as η -► oo.
fcGN
Definition 1.1.3. A sequence {Χη·, η > 1} is said to be φ-mixing от
uniformly strong mixing if
φ(η) = sup^JF^jF^J -+0 as η -^ oo.
fcGN
Definition 1.1.4. A sequence {Χη·, η > 1} is said to be ф-mixing or
*-mixing if
ф{п) = supV^^^J -► 0 as η -^ oo.
fcGN
Definition 1.1.5. A sequence {Xn, η > 1} is said to be absolutely
regular if
β{η) = ΒχιρβίΓΪ,Γ&η) -> 0 as η -н. оо.
fcGN
Definition 1.1.6. Let 0 < α,/3 < 1,α + /3= 1. A sequence {Xn, η >
1} is said to be (α,/З) -mixing if
A(n) = supA(^f,^7^n) —► 0 as η —> oo.
fcGN
Remark 1.1.1. The versions of the above definitions for a sequence
with time-parameter set R+ or i? or Ζ are trivial.
Remark 1.1.2. The concept of α-mixing was introduced by
Rosenblatt (1956). The concept of p-mixing was introduced by Kolmogorov
and Rozanov (1960). Dobrushin (1956) first introduced the definition of
(^-mixing for a Markov process. This definition for a stationary process
was presented by Ibragimov(1959) and Rozanov and Volconski (1959)
respectively (one can also trace back to Hirschfeld 1935 and Gebelein 1941).

1.1 Definitions
5
Absolute regularity was introduced by Kolmogorov (1959),(cf. Rosanov
and Volconski 1959). Blum, Hanson and Koopmans (1963) presented the
concept of ^-mixing, (a, /3)-mixing was introduced by Bradley (1985a) and
Shao( 1989a) independently.
Remark 1.1.3. Doob(1953) showed that a Doeblin irreducible Markov
chain is (^-mixing with φ{η) < abn for some a > 0 and 0 < b < 1;
Rosenblatt (1971) showed that a purely non-deterministic Markov chain is a-
mixing; Davydov (1973) gave a class of Markov chains which are β-mixing.
Remark 1.1.4. For simplicity, we always assume that the mixing
coefficients α(η),ρ{η), · · ·,λ(η) all are non-increasing.
It is clear from the definitions that
p(n) = А1/2д/2(п), λι|0(η) = φ(η) < ψ(η),
and further
ol{ti) < p{n)
by taking X — \a and Υ = 1# in the definition of p-mixing.
Kolmogorov and Rozanov (1960) investigated the relation between a-
mixing and /o-mixing for a Gaussian sequence.
Theorem 1.1.1. For a Gaussian sequence {Χη·, η > 1}, we have
Proof. The former inequality is obvious.
For any ε > 0, there exist two normal random variables X G /^(.Т7^), Υ
G L2{F%.n) such that EX = EY = 0, VarX = УагУ = 1 and
r:=EXY>p{Tl^n)-e.
Noting that A := {X > 0} G T\, Β := {Υ > 0} G Τ^η, we have
P(AB) = - + — arcsinr, P(A)P(B) = - (1.1.1)
4 2π 4
by elementary calculations (see Cramer 1946, p.290). If θί{Τ^^ T^_n) > |,
it is clear that
if α{ΤΊ,^+η) < i, by (1.1.1) we obtain
«(^i.^lb+n) > ^(^B) - ^)P(B) = ^-arcsinr,

6
Chapter 1 Definitions and Basic Inequalities
which implies
Ρ(^ι^^-η) -£<r< sin2na < 2πα.
The theorem is proved by arbitrariness of ε.
Kolmogorov and Rozanov (1960) also studied the relation between the
spectral function of a (weakly) stationary sequence and /9-mixing property.
At first, we give some notations and concepts about a stationary sequence
{Xn, η £ N}. Let the covariance function of {Xn}
R(n) = ЕХшХш+п.
By the Herglotz theorem, there exists the spectral resolution for R(n) as
follows:
R(n)= Г einXdF{\),
J—π
where ^(λ) is called the spectral function of the stationary sequence. When
the spectral function is absolutely continuous, its derivative /(λ) = ^'(λ)
is called the spectral density of the stationary sequence.
Theorem 1.1.2. If the spectral function of a stationary sequence is
not absolutely continuous, then p{n) = 1, i.e. the sequence is not p-mixing.
Conversely, if the spectral function is absolutely continuous, then
p(n) = inf ess sup|/(A) - eiXnh(e-iX)\/f(X),
h χ \ I
where the inf is extended over all functions which is analytically continuable
in unit circle; and further, if there exists an analytic function ho(z) in unit
circle with the boundary value ho(e~lX) such that |/(А)//г-о(в~~гА)| > ε > 0
and (f(X)/ho(e~lX)p ' is bounded uniformly, then
p(n) < cn~
for some с > 0. In particular, when /(λ) is a rational function of егА,
p{n) = e~m
for some с > 0.
The Proof of Theorem 1.1.2 is omitted (Kolmogorov, Rozanov 1960).

1.2 Basic inequalities
7
1.2 Basic inequalities
Let X be -F^ measurable and У be ^+n measurable.
In this section, we establish some bounds of the covariance Cov(X, Y)
EXY — EXEY for the various mixing sequences.
At first, we consider the α-mixing case.
oo
Lemma 1.2.1. Let {Xn,n G Z} be an α-mixing sequence, X G ,F_C
and Υ G T^+n with \X\ < Cx and \Y\ < C2. Then
\EXY - EXEY\ < ACxC2a(n). (1.2.1)
Proof. By the property of conditional expectation, we have
\EXY - EXEY\ = lEiXiEiY]^^) - EY)}\
< CtElEiYlJ*^) - EY\
= 01\Εξ{Ε(χ\1*00)-ΕΥ}\,
where £ = sgn^rlJF^) _ EY) G ft^, i.e.
\EXY - EXEY\ < Οι\ΕξΥ - ΕξΕΥ\.
With the same argument procedure it follows that
\EiY - EiEY\ < 02\Εξη - Ε£Εη\,
where η = sgn (Ε(ξ\Τ^.η) - Εξ). Therefore
\EXY - EXEY\ < СхС2\Е^ - ΕξΕη\. (1.2.2)
Put A = {ξ = 1}, Β = {η = 1}. It is clear that A G J*^ В G F%°+n. Using
the definition of α-mixing, we obtain
\Εϊη-Ε£Εη\
= \P(AB) + P(ACBC) - P(ACB) - P(ABC)
- (P(A) - P(AC))(P(B) - P(B<))\
< 4a(n).
Inserting it into (1.2.2) yields (1.2.1).

8
Chapter 1 Definitions and Basic Inequalities
Lemma 1.2.2. Let {Xn?^ € ^} be an α-mixing sequence, X G
and Υ G J^n тай E\X\* < oo /or some p>\ and \Y\ < С Then
\EXY - EXEY\ < 6C\\X\\p(a(n))1/q, (1.2.3)
where 1/p + l/q = 1.
Proof. Let XN = XI(\X\ < N), X'N = X - XN. Write
\EXY - EXEY\ < \EXNY - EXNEY\ + \EX'NY - EX'NEY\.
By Lemma 1.2.1, \EXNY - EXNEY\ < 4CNa(n). For the second term
of the right hand side of the above inequality, we have
\EX'NY - EX'NEY\ < 2CE\X'N\ < 2CN~P+1E\X\P.
Taking Ν = ||Χ||ρ(α(η))-χ/ρ yields (1.2.3).
For a random variable X and a continuous non-decreasing function
f(x) on R+ with /(0) = 0, which doesn't identically equal to zero, define
\\X\\f = inf{t > 0,Ef(\X\/t) < 1}.
From this definition, it is easy to know that
\\X\\f = 0^1 = 0 a.s. (1.2.4)
and if 0 < \\X\\f < oo, then Ef(\X\/\\X\\f) < 1. Moreover, if \Хг\ < \X2\
a.s., then HJsTxIl/ < ||Х2||/.
Lemma 1.2.3. Let {Χη·> η G Ζ} be an α-mixing sequence, X G
^ΐοο, Υ G ^+n, f(x) and g{x) be two continuous functions on R+
with /(0) = #(0) = 0, f(x)/x~ /* oo and g(x)/x « /" oo for some
r > 0,s > 0,\\X\\f < oo,\\Y\\g < oo. Then
\EXY - EXEY\ < 10inv/(—Ц) mvg(-^-)a(n)\\X\\f\\Y\\g. (1.2.5)
Proof. It is easy to see that E\X\1+s/r < oo and E\Y\1+rls < oo by
the conditions of the lemma . If either ||X||/ = 0 or \\Y\\g = 0, (1.2.4)
implies that (1.2.5) holds. If a(n) = 0, (1.2.5) is trivial by independence of
X and Y. Now we assume that ||X||/ > 0, \\Y\\g > 0 and a(n) > 0. There
are Μ > 0 and TV > 0 such that
a(n) = l/f(M/\\X\\f) = \/g{N/\\Y\\g).

1.2 Basic inequalities
9
Let
XM = XI{\X\ < M), X'M = X-XM,
YN = YI(\Y\<N), Y'N = Y-YN.
We have
\EXY - EXEY\
< \EXmYn - EXMEYN\ + \EX'MYN - EX'MEYN\
+ \EXmYn ~ EXMEY'N\ + \EX'MY'N - EX'MEY'N\
=: h + I2 + I3 + h. (1.2.6)
By Lemma 1.2.1, Ix < 4MNa(n). Noting that f(x)/x / oo and g{x)/x /
oo, we have
E\X'M\ = E(\XM\/\\XM\\f)\\x'M\\f
<Ef(\x'M\/\\XM\\f)M/f(M/\\XM\\f)
< M/f(M/\\X\\f).
Therefore
h < 2MN/f(M/\\X\\f) = 2inv/(^) invg(^)a(n)\\X\\f\\Y\\g.
Similarly, we have the same estimation for /3.
Furthermore, noting that f(x)/x~^~ / 00 and g(x)/x~ / 00, we
have
ex'my'n < (е{\хм\/\\х'мЬ)^)^
•(Bd^l/H^rlW^J^II^II/lll^ll,
■MN/(f(M/\\X'M\\f))^(g(N/\\Y^\\g))^
< MN/(f(M/\\X\\f))^ (g(N/\\Y\\g))^.
Hence
h < 2ΜΝ/(/(Μ/||Χ||/))*(!,(]ν/||Κ||ι,))*
^ata"(55o)te"(5So)°<-)'Ar«''yb·

10
Chapter 1 Definitions and Basic Inequalities
Now, inserting these estimations into (1.2.6) yields (1.2.5).
As some consequences of this lemma, we have
Lemma 1.2.4. Let {Xn? τι G Z} be an α-mixing sequence, X G J7-^
and Υ G T^_n with E\X\* < oo and E\Y\4 < oo, \ + \ < 1. Then
\EXY - EXEY\ < 10\\Χ\\ρ\\Υ\\4(α(η)γ-ρ-<. (1.2.7)
Lemma 1.2.5. Let {Xn, τι G Z} be an α-mixing sequence, X G ^oo
and У G ^n ιΐ7*ίΛ Ε\Χ\2+δ < d,E\Y\2+6 < C2. Then
\EXY - EXEY\ < 10(CiC2)^b(a(n))^. (1.2.8)
For an (a,/3)-mixing sequence and a /o-mixing sequence, we have the
following lemmas.
Lemma 1.2.6. Let {Χη, η G Ζ} be an (a^P)-mixing sequence, X G
lp(P-oo) and Y ^ Lqi^k+n) with P,4>1 and 1/P + 1/9 = 1· Then
\EXY - EXEY\ < 4\(n)^AW-*\\X\\p\\Y\\q. (1.2.9)
Proof. Without loss of generality, assume that ap > 1, which implies
that βς < 1. Put
Fx = YI(\Y\ < С), У2 = У-УЬ
where С is a positive constant specified later on. Write
\EXY - EXEY\ < \ΕΧΥλ - EXEYX\ + \EXY2 - EXEY2\. (1.2.10)
By the definition of (a, /3)-mixing and the Holder inequality
\EXYX - ΕΧΕΥ,Ι < λ(η)||Λ:||1/β||^||1//9
< Цп^-ецхигце*,
\ΕΧΥ2\ < {E\Y2\qY~^(E\X\a^\Y2\^)^
< {Ε\Υ2\«γ-^ (Ε\Χ\αΡΕ\Υ2\β(> + λ(η)(Ε\Χ\ηα(\Υ2\ηβ) "
< (E\Y\qY-^ (E\X\apE\Y\"C-aq + \(η)(Ε\Χ\ρ)α(Ε\Υ\ηβ) ^
<\\x\\p\\Y\\4qc-i + xMn)\\x\\P\\Y\\q

1.2 Basic inequalities 11
and
\exey2\ < \\x\\p\\y\\4c-*'».
Inserting these estimations into (1.2.10) and taking С = \\Y\\q(X(n))~ 'aq
we obtain (1.2.9).
Let ρ = q = 2 in (1.2.9). It is easy to see that
p(n) <4λ(η)^Λ^. (1.2.11)
As a consequence of Lemma 1.2.6, noting that p(n) = X1/2,1/2(^)5 we
have
Lemma 1.2.7. Let {ХП5 η £ Z} be a p-mixing sequence, X G Lp^ll^)
and Υ G Lq(T^_n) with p, q > 1 and 1/p + 1/g = 1. Then
\EXY - EXEY\ < 4р(п)рАя\\Х\\р\\У\\д.
For the (^-mixing case, we have the following three results.
Lemma 1.2.8. Let {ХП5 η G Z} be αφ-mixing sequence, X G Lp(^00)
and Υ G Ι/ρ(^^_η) with p,q > 1 and 1/p + 1/g = 1. Then
\EXY - EXEY\ < 2(φ(ή))ρ\\Χ\\ρ\\Υ\\4. (1.2.12)
Proof. At first, we assume that Xand Fare simple functions, i.e.
X = YiaiIAi, Y = Y/bjIBj,
where both Σί and Σ^ are finite sums and AiDAk = 0 (г ^ fc), BjDBi
0 (j # Z), Ai € ^те, β,- G JT~n. So
EXY - EXEY = Y^aibjPiAiBj) - 53<цЬ,-Р(А*)Р(В,·).

12
Chapter 1 Definitions and Basic Inequalities
By the Holder inequality we have
\EXY-EXEY\
* 3
< (J2\ai\^P(Ai)y/P(j2P(Ai)\Zbj(P(Bj\Ai)-P(Bj))\'iy/q
г г j
<\\Х\\Р\^Р(А{)(£\Ъ1\*(Р(В№{)
* j
+ р(В1)))(52\Р(вМг)-Р(в>)\)1р\<
3
<2V^X\\p\\Y\\qmaxfc\P(Bs\*)-P(Bj)\y/P· (1-2-13)
3
Note that
£|Р(я,-|л<) - p(Bj)\ = {P{^BMi) ~ PiV+Bj))
j
-(PiujBjW-PiujBA)
< 2φ(η), (1.2.14)
where the union U+(UJ) is carried out over j such that P(Bj\Ai)—P(Bj) >
0 (P(Bj\Ai) - P(Bj) < 0). Inserting (1.2.14) into (1.2.13) yields (1.2.12)
for the simple function case.
In order to complete the proof of the lemma, let
(0 if
XN = \k/N if
|X| > N
k/N < X < (k + 1)/N, \X\ < N;
■f
, 0 if \Y\ > N.
л k/N if k/N <Y <(k + 1)/N, \Y\ < N.
We have showed that (1.2.12) is true for Xn and Yff. Moreover, note
E\X-XN\p-^0, E\Y-YN\q -^0, asJV^co.
Letting N —у со, we obtain (1.2.12) for the general case.
Let ρ = q = 2 in (1.2.12). It is easy to see that
p(n) < 2ψχΙ'2{η). (1.2.15)
From the proof of Lemma 1.2.8, we can see that

1.2 Basic inequalities
13
к
-oo
Lemma 1.2.9. Let {Xn, η € Z} be a φ-mixing sequence, X £ T[
and Υ £ ^n with \X\ < Cx and \Y\ < C2. Then
\EXY - EXEY\ < 2C1C2<p(n). (1.2.16)
Let ρ = 1 and q = oo in (1.2.12). From Lemma 1.2.8, we also have
Lemma 1.2.10. Let {Xn? η £ Z} be a φ-mixing sequence, X £ T1!^
and Υ £ JF^n with E\X\ < oo and \Y\ < С Then
\EXY - EXEY\ < 20φ(η)Ε\Χ\. (1.2.17)
Finally, we consider the -^-mixing case.
Lemma 1.2.11. Let {ХП5 η £ Z} be a ψ-mixing sequence, X £ T^.^
and Υ £ Tjfi_n with E\X\ < oo and E\Y\ < oo. Then E\XY\ < oo and
\EXY - EXEY\ < ψ(η)Ε\Χ\Ε\Υ\. (1.2.18)
Proof. At first, we assume that X and Υ are non-negative simple
functions. We have
\EXY - EXEY\ = l^aibjiPiAiBj) - Р{А{)Р{В^))\
<Yjaibj^{n)P{Ai)P(Bj)
ij
= ψ(η)ΕΧΕΥ.
From this, (1.2.18) holds for non-negative random variables X and Y.
For the general case, write X = X+ — X~, Υ = Y+ — Y~. We have
\EXY-EXEY\
< \EX+Y+ - EX+EY+\ + \EX+Y~ - EX+EY~\
+ \EX~Y+ - EX~EY+\ + \EX~Y~ - EX~EY~\
< ψ(η)(ΕΧ+ + EX~){EY+ + EY~)
<ψ(η)Ε\Χ\Ε\Υ\.
Finally, we summarize the relations between one and another of variaous
mixing properties. It is easy to verify that
2a(n) < β(η) < φ(η).
(1.2.19)

14
Chapter 1 Definitions and Basic Inequalities
With a necessary and sufficient condition for Markov processes to be ψ-
mixing, one can show that a (^-mixing (Markov) sequence is not ^-mixing
(Blum, Hanson and Koopmans 1963). Ibragimov and Solev (1969) given an
example of a stationary α-mixing Gaussian process which is not β-mixing;
such a process is /o-mixing but not /3-mixing. Davydov (1973) constructed
a stationary α-mixing Markov process with less than geometric rate of
decay of the mixing coefficients, which is not /o-mixing. It is possible that
a geometrically ergodic Markov process which is not Doeblin recurrent is
/3-mixing and not <£>-mixing (Andrews 1984). Combining these results and
recalling Remark 1.1.4, (1.2.11) and (1.2.15) we have
φ— mixing < \ψ— mixing { { # $
L
β — mixing < >a — mixing
и
ρ— mixing^ / >a
mixing
ft
λ — mixing

Chapter 2 Moment Estimations of Partial Sums
The estimations of some kind of moments of partial sums of a mixing
sequence play important roles in showing limit theorems. In Section 2.1,
we give some forms of the variances of partial sums of mixing sequences
of various kinds. Section 2.2 is devoted to deduce some inequalities for
the moments of partial sums. In passing we also give some probability
inequalities in this section.
2.1 Variances of partial sums
Let {Xn-> η Ε Ζ} be a (weakly) stationary sequence with EX\ =
0, EX\ < oo. Put Sn = Σί=ιΧί- We investigate its variance VarS^. Let
R(n) be the correlation function and F(X) be the spectral function of the
sequence {Xn}·
At first, we give the representations of VarSn by R(n) or F(X).
Theorem 2.1.1.
VarSn= Σ(η-|ίΊ)Λϋ), (2.1.1)
\i\<n
/π sin^ —
ННЖ). (2.1.2)
-π Sill ^
If the spectral function is absolutely continuous, i.e., there is a spectral
density /(λ), and further, if /(λ) is continuous at λ = 0, then
VarSn = 2π/(0)η + o(n) as η-^ oo. (2.1.3)
The proof of the theorem can be found in the book of Ibragimov and
Linnik (1971) and is not presented here.

16
Chapter 2 Moment Estimations of Partial Sums
When a stationary sequence satisfies a certain mixing condition, VarSn
possesses more evident form.
Theorem 2.1.2. Suppose that a stationary sequence {Xn} i>s φ-mixing
and VarSn —> oo as η —> oo. Then
VarSn = nh(n), (2.1.4)
where h(n) is a slowly varying function of η and its domain of definition
can be extended to R such that h(x) is also slowly varying on R.
Proof. We first prove that h{n) is slowly varying. Put σ2 = VarSn.
Equivalently, we show that for every positive integer к
]ima2kn/a2n = k. (2.1.5)
Let
0 — 2s X(j-l)n+(j-l)r+si J — 1? 2, · · ·, ft,
s=l
r
Vj — 2s ^jn+(j-i)r+si J = l)2, ···,& — 1,
s=l
(k-l)r
Vk = ~ 2s Xnk+si
s=l
where r = [loga2]. By Theorem 1.1.2, {Xn} possesses a spectral density
/(λ). Using (2.1.2) we obtain
J-* (sin2|j
<n2 Γ f(X)d\. (2.1.6)
J—π
Hence r = O(logn). And further
к
σ\η = VarSfcn = Σ,Ε$ + 2Σ, Ε^ + Σ Et*H + Σ EWi- i2'1'7)
By stationarity of the sequence, Εξ? = σ2 = VarSn. From Lemma 1.2.8,
we have
\Εξ&\ < M\i-J\r)1/2\M2\\tih < Mr)1/2*l (2-1.8)

2.1 Variances of partial sums
17
for г Ф j. Using the Schwarz inequality and (1.2.6), we have
\E£mj\ < Ц&1ЫЫ12 = σησΓ = 0(anlogan), (2.1.9)
\EtHVj\ < σΐ = 0((bgan)2). (2.1.10)
Inserting (2.1.8), (2.1.9) and (2.1.10) into (2.1.7) and noting ψ{τ) = o(l)
as η —> oo, we obtain
which implies (2.1.5).
Next, we prove that the domain of h(n) can be extended to R such
that h(x) is also slowly varying on R. Recalling (2.1.6), we define
/π oir|2 -ελ
-rf/WA
-π Sin"5 ^
h(x) = φ(χ)/χ.
In order to show that h(x) is slowly varying, it is enough to verify that for
any a > 0
lim φ(αχ)/φ(χ) = a. (2.1.11)
x—»oo
It is not difficult to know from the definition of φ that
ψ(χ) = ψ([χ])(ΐ + o(l))
as χ —> oo. When α in (2.1.11) is an integer, we have
7Й" = ИМИ) ( + ( )} = ( + о(1))·
Therefore, for a — p/q, where both ρ and q are integers, we obtain
Um ^M = lim ^f) ^f) = Ρ =
™ φ(χ) x^L Щ) ф(д^) g
For any positive real number a, put
For any rational number а, V'lW = ^2 (a) by the above proof. Hence, it
suffices to show that both ψι(χ) and -02(x) are continuous. Because
ι ^((^ + ε)#) — φ (ax) I
1 ι /*π sin2^ _ч „ Г sinexAsinaxA
<
Г sin^ . . ιλ г* sinexAsinaxA г/лч ,л
7-π siir £ J-π 2siir£
V>(x)
φ(εχ
φ(χ) ^ V^1)
2 "_л лоххх 2

18
Chapter 2 Moment Estimations of Partial Sums
it is enough to show that φ\{α) and -02(a) axe continuous at a = 0. Using
Property A4 about a slowly varying function(see Appendix A), for ε > 0
small enough, we have
φ(εχ) [ex]h($[x])
Φ) [x]h([x]) [i + °[i))
<e1/2(l + o(l))
as χ —> oo. Hence both ψ\{α) and V>2(a) are continuous at a = 0. Theorem
2.1.2 are proved.
Remark 2.1.1. In the proof of Theorem 2.1.2, (^-mixing property is
only used to give the inequality
m+n+p
E\sn Σ *i|<2<Kp)1/2||Sn||2||STO||2.
j=n+p
Then, for α-mixing case, we have also
Theorem 2.1.3. Let {Xn} be a strictly stationary α-mixing sequence
satisfying that ΕΧχ = 0, EX\ < oo, σ£ = ES% —► oo and {5^/σ^, η > 1}
is integrable uniformly. Then the conclusions of Theorem 2.1.2 hold true.
Proof. By the proof of Theorem 2.1.2 and Remark 2.1.1, it suffices
to show the following facts.
1. al -> oo;
2. for any ε > 0, there exist ρ = ρ(ε), Ν = Ν (ε) such that
m+n+p
ESn Σ XJ\^
εο'ηο'πι if η, m > 7V(s).
j=n+p
The first fact is an assumption of the theorem. Consider the latter.
From uniform integrability of {S^/σ^}, for any ε > 0, there exists а К > 0
such that for ρ large enough,
/.
S'JaidP<- Ka(p)< ε/16.
Then, by Lemma 1.2.1, Schwarz's inequality and strict stationarity, we

2.1 Variances of partial sums
19
obtain
\ESn 22 Xj\/σησΏ
m+n+p
Σ
j=n+p
<
J\§*\<
I c
ι ι στη '
m+n+p ^n+p—1
Cm
dP
+ / , v^| — \dP
Ι ση l —
>n
m+n-p ^n+p—1 |>ν/]7
Vk
^m+n+p
bn
+P-1
dP
η °m+n+p ~ ^n+p—1
dP
l\s*\>y/K,\Sm+n+T s"+p-'|>7k|
Ι ση ' — ' στη ' ~
< 4Κα(ρ) + ε/4 + ε/4 + ε/4 < ε.
For a /o-mixing sequence, Peligrad (1982) showed the following general
result. Denote Sk(n) = Sk+n - Sk = Ejijb+i Xl·
Theorem 2.1.4. Let {Xn, η > 1} be a p-mixing sequence of random
variables with EXn = 0. Assume that
(i) supn EXl = al < oo;
("n^ .E5^ —► oo as η —> oo;
..... .. £Sjg(n) l ., _ . .
(шу limn_^oo — * = 1 uniformly in k.
&bn
Then
ESl = nfc(n),
where h(n) is a slowly varying function and its domain of definition can be
extended to R such that h{x) is also slowly varying on R. If, in addition,
assume that
(iv) Σ^=ιΡ(2η)<οο,
then ESl/n -> σ2 > 0.
In order to prove Theorem 2.1.4, we need the following lemma.
Lemma 2.1.1. Let {ХП5 η > 1} be a p-mixing sequence with EXn = 0.
// condition (i) in Theorem 2.1.4 i>s satisfied, then, for natural numbers
p, g, m with p + q = m,
(1 - p(n))(ESlm(p) + ES2km+p{q)) - d
< ESlm{m)
< (1 + p(n))(ES2km(p) + ESlm+p(q)) + d, (2.1.12)

20
Chapter 2 Moment Estimations of Partial Sums
where к and η are positive integers and
d = Ci(m,p,n) < 20σ*η2 + l2a0n(\\Skrn(p)\\2 + ||S*m+p(9)||2),
and further
(1 - p(n))1/2\\Skm(p)\\2 < \\Skm(m)\\2 + C2 (2.1.13)
where C2 < 2σοη.
Proof. By the definition of p-mixing, we have
(q))2-(ESlm(p) + ES2km+p+i(q))\
< p(i)(ESlm(p) + ES2km+p+i(q)). (2.1.14)
Noting that Skm+p+n(q) = Skm+p(q) - Skm+P(n) + 5(fc+1)m(n), we obtain
\\Skm+p+n(q)\\2 = 115^^(9)112+^, (2.1.15)
where |0i| < 2σοη. Hence, from (2.1.14) and (2.1.15), it follows that
(1 - p(n))(ESlm(p) + ES2km+p(q) + 02)
< E(Skm(p) + Skm+p+n(q))
< (1 + p(n))(ES2km(p) + ES2km+p(q) + θ2), (2.1.16)
where \θ2\ < 4σ^η2 + 4a0n\\Skm+p(q)\\2. Write
Skm(m) = Skm(p) + Skm+P(n) + Skm+p+n(q) - 5,(fc+1)m(n).
Then
||5fcm(m)||2 = \\Skm(p) + Skm+p+n(q)\\2 + θ3, (2.1.17)
where |03| < 2σ0η. Hence ESlm(m) = E(Skm(p) + Skm+p+n(q))2 + θ4,
where
\θ4\ < ΥΙσΙη2 +4a0n(||5fcm(p)||2 + ||5fcm+p(9)||2).
Inserting it into (2.1.16) we obtain (2.1.12), where
d = max(|(l - ρ{η))θ2 + ΘΛ\, |(1 + ρ{η))θ2 + θ4\)
< ΊΟσΙ,η2 + 12a0n(\\Skm(p)\\2 + \\Skm+p(q)\\2).
We turn to (2.1.13). (2.1.14) implies
(1 - p(n))ES2m(p) < E(Skm(p) + Skm+p+n(q))2.
Then (2.1.13) is showed from (2.1.17).

2.1 Variances of partial sums
21
Proof of Theorem 2.1.4. At first, we prove that for any h £ N
\imQES2hJESl = h. (2.1.18)
By (2.1.12) with к = 0, m = hn, ρ = (h - l)n, q = η, η = [(SS*)1/3], we
have
(1 - p(n))(ESfh_l)n + ESfh_1)n(n)) - Co
<ES2hn
< (1 + p(n))(JESffc_1)n + £S(Vi)») + Co,
where C0 = 20σ^η2 + 12σοη(||5(/ι_1)η||2 + ||5(Λ-ι)η(«)Ι|2)· Using conditions
(ii) and (iii), (2.1.18) follows by induction on h. Therefore,
h{n) := ES2Jn
is a slowly varying function. Extend its domain by letting
h(t) = ESft]/t.
We show that
lim Λ((1 - en)n)/h(n) = 1, (2.1.19)
n—►oo
where εη j 0 suah that nen are integers. By Property A4 (in Appendix),
lim btg£e6 = 0u (2ЛЩ
П-+00 h(n) П V '
Let hn = max(/i : /ιηεη < (1 — εη)η). Note that
^n-nen = &n ~ ^hnnen\n£n) + ^hnnen \P) ~ ^hnnen-\-nen\P)ι
where ρ = η — (hn + 1)ηεη < ηεη. By (2.1.13) with га = ηεη, к = hn and
/ιη + 1, we have
I Sn-n£n || 2 — || Sn\\21
< ΙΙ5/ΐηηεη(^εη)||2 +
(l-p(i))V2
+ l|S(/in+i)nen(^n)||2 +4a0i),
(ΙΙ^ηηεη(ηεη)||2
where г is such that р(г) < 1. Dividing both sides of the above inquality by
||5n||2 and using (iii) and (2.1.20), we obtain (2.1.19). For integer к > 0,
there exists n& such that for every η > η&,
log
h{nk) ι
h(n) I
<
1
(2.1.21)

22
Chapter 2 Moment Estimations of Partial Sums
since h(n) is slowly varying. Without loss of generality, assume that n^ is
strictly increasing on k. Let t > 1. For integer η > 0, define к = qn such
that rik < nt < rifc+i. Then, (2.1.21) implies that
lim log(h([nt]qn)/h(nt)) = 0. (2.1.22)
Put pn = [qnt]. Then pn = [kt] > к. Hence, (2.1.21) also implies that
lim log(h(npn)/h(n)) = 0. (2.1.23)
n—► oo
Moreover, from (2.1.19) we have
lim h([nt]qn)/h(npn) = 1.
71—► OO
Combining it with (2.1.22) and (2.1.23) yields
lim h(nt)/h(n) = 1.
η—►oo
Therefore, by Property Al of Appendix
1# h(xt) л. h([x]t)
lim -L^ = lim -АШ = ι
ж-юо h[x) χ-+°° h([x\)
as required.
Now we consider the second part of the theorem. By (2.1.12) with
m = 2ΛΓ, p = q = N, n= [JV1/3], we have
(l-pdAT1^]))^^^) + £;5(22fc+1)iV(^))(l - α*)
< ES22kN{2N)
< (1 + р([ЛГ1/3]))(£;522^(ЛГ) + BS(22fc+1)N(JV))(l + а„*2.1.24)
where
20σ02ΛΓ2/3 + i2q02iVi/3(||52fcjV(iV)||2 + ||5(2fc+1)Jv(iV)||2)
aN Τ (1 - p([N^)))(ESlkN(ЛГ) + SS22fc+1)„(iV))
and iVo is so large that p([JVx/3]) < 1 for N > N0. Conditions (i) and (Hi)
imply that
'tfW + NWWSsh*
aN - ■ ·■
~ V IL9vl|2 )'
11ЗД
By Property A2 of Appendix, for any 0 < ε < 1/6
lim N£ESl/N = oo.
N—>oo

2.1 Variances of partial sums
23
Hence (ESjf)-1 = 0(Ν~1+ε), and further,
aN = 0(Ν-*+ε). (2.1.25)
Then (2.1.24) implies that for integers r>p>N0 with p([2N°/3]) < 1,
Π(1-ρ([2*/3]))(1-α20 Σ Я^(2Р)
г=р г=0
< Εθ2τ
<П(1 + р([2^3]))(1+а20 ^ ESf2P(2p). (2.1.26)
г=р г=0
By conditon (iv), Σι Ρ([2*/3]) < °°· Moreover, (2.1.25) implies that Σι α(2')
< οο. Therefore, from (2.1.26) we obtain
r^oaEs^fiiEs^2P)=i·
Consequently, it follows by condition (iii) that
r:HmeoM2P)A(2P) = l,
and further, h(2r) converges to a positive constant. Applying Property A3
of Appendix to h(t) and l//i(£), we obtain that h(n) converges to the same
limit as h(2r). Theorem 2.1.4 is proved.
For a strictly stationary p-mixing sequence, we also have the following
result.
Theorem 2.1.5.(Ibragimov 1975) Let {Xn, η > 1} be a strictly
stationary p-mixing sequence with EX\ = 0, EX\ < oo and Σ^=ι p(2n) < oo.
Then {Xn} possesses a continuous spectral density /(λ) and
VarSn = 2π/(0)η + o(n) as η —> oo
.//(0)^0.
For the proof of Theorem 2.1.5 we refer to Lemma 17 of Ibragimov and
Rozanov (1978). They first show that {Xn} possesses a bounded spectral
density if Σ™=ι P(2n) < oo, and then show that
oo
En(f) < 128 max/(A) J£p(2k~1n),
л fc=o

24
Chapter 2 Moment Estimations of Partial Sums
where En(f) denotes the error of best approximation of /(λ) by a
trigonometric polynomial of degree less than or equal to η on [—π, π]. With the
help of this result, we may know that /(λ) is a continuous function on
[—π, π]. The rest of the proof can be completed by using Theorem 2.1.1.
2.2 Further inequalities
In order to show limit theorems for a mixing sequence, we often need
some further inequalities besides the basic inequalities in Section 1.2.
The following extended Ottaviani inequality for an α-mixing sequence
was given by Lin (1982).
Recall the notation T\ — σ{Χ%, α < г < b) for a sequence {Xn} of
random variables.
Lemma 2.2.1. Let {Xn,n > 1} be an α-mixing sequence. For any
given integers p, q and /с, let £j be ^SuL ui measurable, j = 1,2, · · ·, k.
If
^{|6+i + · · · + Ы < C} > |, / = l, · · ·, к - ι,
then
Pimax |ft + ... + u| > 2C) <2Р{\^ + ---+£к\>С} + 2ка(д).
Proof. Let events
A = { max |& + ... + 6| > 2CJ, В = {|6 + · · · + &| > C},
Al = {\£1\>2C},
At = { max |£1 + ... + £r|<2C, \ξι + ■ ■ ■ + ξι\ > 2C\, l = 2,---,k,
4<r<i—1 J
в* = {|&+i + ···+ &|<σ}, ι = ι,··-,fc-1, вк = п.
Then
к к
МА^Ъ{гфз\ A=\jAh \jAiBtCB.
1=1 1=1
By the conditions of the lemma,
Ρ(Λ,β,) > P(Ai)P(Bt) - a(q) > ^P(Ai) - a(q),

2.2 Further inequalities
25
and hence
к к
Р(В) > ΣP{AiB{) >\ΣΡ(Αι) - ka{q)
1=1 1=1
= \Ρ{Α) - ka(q)
as required.
The following lemmas all are about the bounds of the moments of
partial sums. For a p-mixing sequence, earlier work was due to Peligrad
(1982, 1987). Shao (1988b, 1989a,b) improved and generalized her results.
Lemma 2.2.2. Let {Xn,n > 1} be a p-mixing sequence with EXn —
0, EX% < oo for each η > 1. Then for any ε > 0, there exists а С —
C{e) > 0 such that
[log n]
ES2k{n) < Cnexp{(l + ε) £ р(2г)} max SX*
г=0 ~~
for each к > 1 and η > 1, where Sk(n) = Σ?=ί&+ι Xi-
Proof. Without loss of generality, assume 0 < ε < \. Let Cn be a
non-decreasing sequence of numbers such that
ESlin) < Cnn max EX?. (2.2.1)
к<г<к+п
For η < 21/6 we need only to take Cn > 2xle by the Minkowski inequality.
Let С χ = 21/ε. We suppose that Cm, m = 1, · · ·, n — 1, are already defined as
demanded in the lemma. Put n\ — [n/2], n2 = η — ηχ, ?гз = [n1^1"1"6)] + 1.
It is clear that
ESl(n) = ESUm) + ES2k+ni(n2) + 2ESk(ni)Sk+ni(n2), (2.2.2)
and further
|£Sfc(ni)Sfc+ni(n2)|
< |SSfc(rai)Sfc+ni(ra3)| + |^5fc(ni)5fc+ni+n3(n2 - ra3)|
<ΙΙ^(ηι)||2||5Λ+ηι(η3)||2
+ p(n3)||5fc(ni)||2|| (n2 -n3)||2
<2\\Sk(n1)\\2\\Sk+ni(n3)\\2 + p(n3)\\Sk(n1)\\2\\Sk+ni(n2)\\2.

26
Chapter 2 Moment Estimations of Partial Sums
Inserting the above inequality into (2.2.2) and noting (2.2.1) we obtain
ESl(n) < (ESKm) + ES2k+ni(n2))(l + p(n3))
+ 4||5fc(n1)||2||5fc+ni(n3)||2
- -
< Cn2 · ra(l + p(n3)) max EX? + 4Cnin12n32 max EX?
k<i<k+n k<i<k+n
<Cn2(l + p(n21+£) + 4n22(1+£))n max EX?,
k<i<k-\-n
where
p(x) = (p(i + 1) - p(i))(x - i) + p(i) if г < χ < г + 1.
Hence for η > 2, we define
Cn = СП2(\ + р{пр) + 4^^).
Obviously Cn is nondecreasing, and
C2» = C2n-i(1 + ρ(2^τ) + 4 · 2~^+ϋ)
n-l
= Ci Π(ΐ + ρ(2ΐ^) + 4·2 2(i;e)j
i=0
n-l £i
< Ciexp{5^(p(2bfe) + 4-2~2^)}
i=o
<CiexpJ3+ / p(2^)dx + c£}
< Сг exp{3 + (1 + ε) Γ p(2x) dx + c£\
L ^2/(l+e) J
n-l
< d exp{3 + (1 + ε) £ p(2*) + c£}, (2.2.3)
i=l
where c£ = 4/(1 - 2~ε/(2(1+ε))). Put de = 21/£exp(3 + c£). We get
n-l
C2n<deexp{(l + e)Ytp(2i)}.
i=l
For any n, there exists an m such that 2m < η < 2m+1. Using the mono-
tonicity of Cn, it follows that
m
Cn < C2m+i < deexp{(l + e)5^p(2<)}
[log n]
<4exp{(l + e) Σ Р&)}-
i=0

2.2 Further inequalities
27
The lemma is proved.
Lemma 2.2.3. Let {Xn,n > 1} be a p-mixing sequence with EXn =
0,ЕХ% < ос for each η > 1. Suppose that
ESl(n)/ min EXf-^oo as η -> oo (2.2.4)
fc<z<fc+n
uniformly in к and
max £X2 < α min £X2 for some a > 1. (2.2.5)
fc<z<fc+n fc<z<fc+n
■ч,' ^-ч,' >
Then, for any ε > О, Йеге еш! С = С (ε, ρ(·), α) > 0 and an integer N
such that for each к > 0 and η > N
[log n]
ESl(n) > 6''nexp{-(l + ε)Σ /°(2')} , ™? Я*?·
^ T~T J к<г<к+п
г=0 *
Proof. Without loss of generality assume that 0 < ε < 1/400.
Consequently,
1 - 5ε2 > (3/2)~ε/6.
Hence, noting p(n) —> 0 as η —► oo, we have
1 _ 5ε2 _ p(mo) > (3/2)-«/e
for some large mo. It is not hard to verify that exp{2^^Q р(2г)} is a
slowly varying function. Then, by Lemma 2.2.2 and condition (2.2.4),
there exists an no such that for η > no
ESl(n) < η1+ε max EXf, (2.2.6)
k<i<k+n
ESl(n)>^p. min EXf. (2.2.7)
K ' ~ ε4 k<i<k+n l v '

28
Chapter 2 Moment Estimations of Partial Sums
When η > 2no, put ri\ — [n/2], ri2 = η — щ. Then
ES2(n) = ES2k(m) + ES2k+ni(n2)
+ 2ESk(n1)Sk+ni(m0) + 2ESk(ni)Sk+ni+mo(n2 - m0)
> ESKm) + ES2k+ni(n2) - 2||5fc(nx)||2||5fc+Tll(mo)||2
-2p(m0)||5fc(n1)||2||5 (n2 -m0)||2
> (1 - p(mo))(ES2k(ni) + ES2k+ni(n2))
-4||5fc(ni)||2||5fc+ni(mo)||2
> (1 - pimoWESKm) + ES2k+ni(n2)) - 4e2ES2(ni)
4 о 2
ъШп max EXf
ε2 k<i<k+n
> (1 - 4ε2 - p(m0))(£;52(n1) + £S2+ni (n2))
4
rrana min £?X,2
ε2 fc<i<fc+n
> (1 - 5ε2 - pim^ESlin,) + £Sfc2+n>2))
> (З/^-^С^СпО + ^+пДпз)). (2.2.8)
We first show that for each n> щ
ESl(n) > C2nl-£'& min EX?, (2.2.9)
k<i<k+n
where C2 = 2ат1щ1£-А. By (2.2.7), (2.2.9) holds for n0 < n< 2η0. When
η > 2ηο, we assume that (2.2.9) is true for each positive integer less than
n. Then it is also true for n. In fact, using (2.2.8) we obtain
ESlin) > (3/2)-£/6C2(n}-£/6 + n^e/6) min SX?
fc<z<fc+n
•η1"6/6 min ЯХ?
fc<z<fc+n
> C2nl~£l& min EXf
as required.
Next, we turn to the assertion of the lemma. For η > riQ+e, put
щ = [§], n2 - η - щ, n3 = [n1/^)] + 1. From (2.2.6) and (2.2.9), it

2.2 Further inequalities
29
follows that
ES2k(n)
= ES2k{ni) + ES2k+ni(n2) + 2ESk(ni)ESk+ni(n3)
+ 2ESk(ni)Sk+m+n3(n2 - ra3)
> ES2{nx) + ESl+m(n2) - 4||5fc(n1)||2||5fc+ni(n3)||2
-2p(n3)||5fc(n1)||2||5fc+ni(n2)||2
> (1 - p(m))(ES2k(ni) + £Sfc2+ni(n2))
- 4(гащ3)(1+е4)/2 max EX?
к<г<к+п
> (1 - p(n3))(SSf (щ) + SSf+ni(ra2))
_ 4nl-(e-3e^)/2(l+e) max £χ2
к<г<к+п
> (1 - p(n3))(SSf (»n) + SS2+rn (ra2)) - 4ara1~e/5 fc<mm+n EX2
> (1 - p(n3))(ESl(m) + ES2k+ni(n2)) - SaC^n-'^ESKm)
> (1 - p(n3) - n-e/40)(£52(rax) + ES2k+m(n2)). (2.2.10)
The last inequality holds for η > (ваС^1)120/*. Put
n^ = max(ra*+e, (ваС^1)120/6).
Let Cn be non-increasing so that
ESl(n) > Cnn min EX?
к<г<к+п
for η > ra0. Then, by (2.2.10)
ES2(n) > (1 - p(ra3) - П2-е/40)СП2п min £X?
к<г<к+п
> (1 - p(ra2^) - ra2-e/4°)Cn2ra mm ЯХ2
«<г<«+п
for η > n'0. Hence we can choose
Cn = Cn2 (l - p(rap) - гаГ/4°) (2.2.11)
for η > n'0. It is easy to see that there exists an η$ such that for η > η$
1 - ρ{η^) ~ n2_e/4° > exp{-(l + ε)(ρ{τφ) +η~ε/40)}. (2.2.12)

30
Chapter 2 Moment Estimations of Partial Sums
Put Пд — n'0VnfQ. In view of (2.2.7), we take Cn* = 4απιΙ/(ε4η^). Obviously
{Cn, η > 2tiq} defined by (2.2.11) is non-increasing. From (2.2.11) and
(2.2.12), we obtain that for 2m > n£
C2m = C2m-l(l - /9(2^) - 2-(m~1)e/40)
> Cam-i exp{-(l + e)(p(2^r) + 2~(m~ 1)ε/40)},
which implies that
m— 1
C2™ > Cn; Π «Φ{-(1 + ε)(ρ(2ί/(1+ε)) + 2--/40)}
i=0
m—1
= CK exp{-(l + ε) £ (р(2^1+£)) + 2"-/40)}.
i=0
Similarly to (2.2.3), there exists a d£ > 0 such that
m—1
/ft X
exp{-(l + ε) £ (/9(2^1+£)) + 2~-/40)}
m
>4ехр{-(1 + б)2^р(2^)}.
г=0
Therefore
m
C2m >4Сп;ехр{-(1 + б)2^р(2^)}.
For arbitrary η > rig, there exists an ra such that 2m < η < 2m+1. By
monotonicity of Cn, we find that
m+l
+i > d£CK exp{-(l + ε)2 £>(2*)}
г=0
[log η]
>-d£Cn;exp{-(l + e)2 £ ρ(2<)},
г=0
and hence we arrive at the assertion of the lemma.
Sometimes we need the bounds of moments of higher than two orders.
Lemma 2.2.4. Let {Xn5 ^i > 1} be a p-mixing sequence with
EXn = 0, supn Ε\Χη\2+δ < oo for some 0 < δ < 1 and
oo
^p(2n)<oo. (2.2.13)
n=l

2.2 Further inequalities
Then there exists а С — C(<5, p(·)) > 0 such that for each η > 1
supE\Sk(n)\2+6 <c{n1+6/2(supEX2)1+6/2
k>l L k>l
+ nexp{(C\ogn)6/(2+V} supE\Xk\2+6}.
Proof. It is not difficult to verify that
(1 + χ)2+δ < 1 + (2 + δ)2(χ + χ1+δ) + χ2+δ
< 1 + 9(χ + ж1+6) + ж2+6 (2
for ж > 0. Put
am = sup ||5fc(m)||2+6, σ™ = sup ||Sfc(ra)||2.
fc>i fc>i
Obviously,
||Sfc(2m)||2+6 < ||5fc(m) + 5fc+m+[ml/5](m)||2+, + 2m1/5a1.
By (2.2.14), putting m\—m-\- [m1/5], we have
E\Sk(m) + Sk+mi(m)\2+6 < 2a2+6 + 9E\Sk(m)\1+6\Sk+rni(m)\
+ 9#|Sfc(m)||Sfc+m^m)|1+*.
Moreover, by the Schwarz inequality and Lemma 1.2.7, we have
E\Sk(m)\1+6\Sk+mi(m)\
< \\Sk(m)\\s2+s\\Sk(m)Sk+mi (m)||(2+6)/2
<α^{σ^ + 4ρ(μ^])α^}2/(2+δ)
<а1а1 + 4рУ^([т^))а^.
Similarly
E\Sk(rn)\\Sk+mi(m)\i+s < alcl + ApW+li\[m^])a2+8.
Combining these inequalities yields that
E\Sk{m) + Sk+mi(m)\2+S
< 2a%s + 18(asmal + 4р^^([т1/«>])<#*)
<
{[2(1 + Збр^+^ат1/*]))] х/(2+«)ат + 18(Tf
Ί2+δ

32
Chapter 2 Moment Estimations of Partial Sums
which implies that
{2(1 + mP^+6\[m^]))}m+8)am + 18am + 2m^ax. (2.2.15)
Noting monotonecity of p(n) and condition (2.2.13), we have
p{n) < c/logn,
here, and in the sequel, с stands for a positive constant, which may take
different values at different places. Hence, applying Lemma 2.2.2, we
obtain
«2- < {2(l + 36p2/(2+fi)([2(r-1)/5]))}1/(2+5)a2.-1
+ 18σ2.-ι + 2 · 2(r_1)/5ai
< г^/^Пи + 36ρ2/(2+δ)([2ί/5]))1/(2+δ)αι
i=0
ι—1 г—1
1/(2+6)
Σ?'2 J] {2(1 + 9р2«2+6\[У/5}))}
+ 2в1Х;2*/5 Π {2(1 + 9ρ2/(2+δ)([2^5]))}1/(2+δ)
г=0 j=i+l
< С2г/2аг + 2r«2+^ exp{Cr)^2+eW (2.2.16)
This implies the conclusion of the lemma.
Similarly, by finer estimation, Shao (1989a) showed the following
results, whose proof will not be presented here.
Lemma 2.2.5. Let {ХП5 τι > 1} be a p-mixing sequence with
EXn = 0, s\ipE\Xn\2+6 < oo for some δ > 0.
η
Then, for any ε > 0, there exists а С = C(<5, ρ(·), ε) > 0, such that for
each η > 2
[1°gn] - · 1+Й/2
E\Sk(n)\2+6<c{(nexp{(l + e) £ ptf)} fc<max+ JX2)
г=0 ~
[log n]
+ nexp{<7 У) р2^2+6\21)} max E\Xi\2+s}
i=0

2.2 Further inequalities
33
Lemma 2.2.6. Let {Xn, η > 1} be a p-тгхгпд sequence with
EXn = 0, E\Xn\q < oo, q > 2, ESl(n) < nh(n) max £Jff.
k<i<k+n
Suppose that there exists a function h(n) such that for every к > 0, η > 1
and there exist a positive integer no and a constant 0 < θ < 21~2^qA3^ such
that
тах(Л([п/2]), h(n - [n/2])) < вк(п)
for η > uq. Furthermore, when q > 3 assume that there exists а С > 0
such that
[logn]
/»(n)>-exp{-C £ ρ2'^)}.
i=0
Then there exists a constant К = K(q, щ, θ, С, р(·)), such that for every
к>0,п>1
E\Sk(n)\q < K\(nh(n) max EX?)
{ k<i<k+n
q/2
[log η
г=0
Next, we turn our attention to a (^-mixing sequence. Peligrad (1985)
showed the following inequality of tail probability (see Shao 1988a).
Lemma 2.2.7. Let {ХП5 η > 1} be а φ-mixing sequence, 0 < η < 1.
Suppose that there exists an integer p, 1 < ρ < η, a number A > 0 such
that
φ{ρ) + max P{\Sn - S{\ > A} < η. (2.2.17)
p<i<n
Then, for any a > 0, b > 0, we have
P\ max \Si\ > a + A + b\
{ l<i<n >
< j^-PUSnl >a} + -l-p{ та* \X<\ > -Ц-}. (2.2.18)
1 — η 1 — ?7 t-iKiKn p—\>
P{\Sn\ >a+A + b}
< ηΡ{ max ISA > a\ + p( max \XA > -). (2.2.19)
~ ίΐ<Κη' ' "" J ll<«<n' ' _ n) v '

34
Chapter 2 Moment Estimations of Partial Sums
Proof. Put Ei = {maxi<7<i \Sj\ < a + A + b < \Si\}. Then
P{ max ISA > a + A + b)
l<i<n
n-l
г=1
n-l
г=1
η—ρ—1
P(|5n| > α) + Σ Ρ(£» П ίΙ5« - $1 ^ Α + *>})
i=l
£р(я<П{|5п-$|>Л + Ь})
п—р—1
< Σ ^(^n{|Si+p-i-Si|>b})
i=l
η—ρ—1
+ £ p(#i η {|sn-$+„_!!> л})
г=1
n-l
+ £ Р(£?<П{|5П-5<|>Л + Ь})
г=тг—ρ
<YtP(Ein{rn^\Xj\>^-i})
г=1
η—ρ—1
+ £ P(i?i)(P{|5n-Si+p-i|>A} + V(p))
< Ρ{ max \ΧΑ > ) + ηΡ\ max \Si\ > a + A + £>},
n<j<n j9 — 1 J ll<t<n J
where condition (2.2.17) is used in the last inequality. Consequently,
(2.2.18) is proved.
As for (2.2.19), putting E{ = {maxi<J<2· \Sj\ < a < |5г|} and noting
\Sn - Sy+p-il > ||5n| - |5y_i| -p max |Xi||forl <j<n-p,
1\г<тг
we have
P{\Sn\ > a + A + b}
< P\\Sn\ > a + A + b, max |5t-| > a, max |X;| < -}
^ 1<г<тг—ρ 1<г<тг ρ)
bj
+ P{max \Х{\ > -}
u<i<n' ' p)
< £ Ρ{Ε[ η {\Sn - Si+p-χΙ > A}) + P{ max |X,| > -}
г=1 г
< ηΡ{ max |5г| > α} + ρ{ max |Хг| > -)
11<г<п > У\<г<п р>

2.2 Further inequalities
35
as required.
Lemma 2.2.8 is due to Shao and Lu (1986).
Lemma 2.2.8. Let {Xn, η > 1} be a φ-mixing sequence with EXn = 0
and supn E\Xn\2+6 < oo for some δ > 0. Suppose that
supESl(n) < MnsupEXl for someM > 0. (2.2.20)
к к
Then there exists а С = C(6, Μ, φ(·)) > 0 such that for each η > 1
supE\Sk(n)\2+6 < Cn1+6/2 supE\Xk\2+6.
к к
Proof. It is easy to see that for r > 1 and χ > 0
(ι+x)r < Σ (1 )*fc+^жГ> (2·2·21)
where 6r = 1 if r is not an integer, otherwise 6r — 0. We now prove
the lemma by induction on r := 2 + δ. Assume that the lemma holds for
/ < [r],r being non-integer. Denoting am = supfc ||Sfc(rn)||r, from (2.2.21)
we obtain
E\Sk(m) + Sk+rn+ko(m)\r
< E\Sk(m)\r + E\Sk+m+ko(m)\r
+ Ejii ( ] ) E\Sk(m)\j\Skwk0(m)rJ
<(2+2ς'ϊ, (;W))< (2·2'22'
+ Σ$ίι f J J E\Sk(m)\jE\Sk+m+k0(m)rJ
=:/i+/2.
By the induction hypothesis, we have
h < Σ{ίχ ( J ) (£;|5fc(m)|H)i/H(£;|5fc+m+fco(m)|H)(-i)/H
< {mW2supkE\Xk\W)r/[r] < cm^^.
Substituting the above inequality into (2.2.22), we obtain
[r] / \
«2m < (2 + 2 JT ί J J ^/Г(^))1/Г«ж+Ст1/2<11.

36
Chapter 2 Moment Estimations of Partial Sums
Now choosing a sufficiently large ко and proceeding as in the proof of
Lemma 2.2.4, we conclude that the lemma holds in this case and similarly
we have the lemma for [r] + 1. This proves the lemma.
Using Lemma 2.2.7, Shao (1988a) proved the following Lemma.
Lemma 2.2.9. Let {Χη? η > 1} be a φ-mixing sequence satisfying
(2.2.17) and q > 0 satisfying ηΑ4 < 1 - η. Then
Ε max \Si\q < (1 - η - η^)-1{{8ΑΥ + 2(4p)qE max \Xi\q\,
l<i<n l 1<г<п J
where 77, p, A are defined in Lemma 2.2.7.
Proof. By Lemma 2.2.7, we have for χ > 8A
P< max ISiI > x\
<
<
Hence for any В > 8A
rB
[ gy^pi max \Si\ > у] dy
JO 4<t<n J
< / gy^PJmax \Si\ > y\ dy
JO 4<t<n J
+ rh qyq-lp{^^\Si\>l}dy
1 — η JsA 4<г<п 4 >
+ T—„ ГяУ^рЫ™ Ш > j~}dy
1 — V JSA U<t<n 4pJ
< (8A)" + -5-4« / qy^Pi max |5<| > y} dy
1 — ?7 Jo 4<t<n ^
+ т^ ГяУч~1Р{ max pb| > y} dy.
1 — TJ JO U<t<n J
which implies that
jf g2/rlP{max |Si|>y}dy
< (1 - η - η^)-1 ((8A)q + 2(4p)qE max |Х;|9У
V 1<г<тг /

2.2 Further inequalities
37
Letting В —> oo yields the assertion of the lemma.
A similar result is
Lemma 2.2.10. Let {ХП5 η > 1} be a φ-mixing sequence. Suppose
that there exists an array {ckn} of positive numbers such that
max ESlii) < ckn. (2.2.23)
1<г<п
Then, for any q > 2, there exists а С = C(q,ip(·)) such that
Ε max \Sk(i)\* < c(cq£ + Ε max \Χ{\Λ. (2.2.24)
1<г<п \ к<г<к+п '
Proof. Take η = 4~2<7, A2 = 2ckn/η.Theτe exists a po such that
ψ{ρο) < ^/2 since φ(ρ) —> 0 as ρ —> oo. Using (2.2.23) we can verify that
(2.2.17) is satisfied. Hence, we get (2.2.24) from Lemma 2.2.9.

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Part II Weak Convergence
In this part, we investigate weak convergence of probability measures
(or distributions ) of normalized sums of the form
±-Σχό-Αη. (Ill)
Bn j=x
We have found a series of this kind of results for an independent sequence
(cf. e.g. Petrov 1971 and Billingsley 1968). A natural question is what
about a weakly dependent sequence. Only assumption of weak dependence
is not enough for weak convergence. For instance, let {£n, η > 1} be a
sequence of i.i.d. random variables with a common characteristic function
/(£), and let
-^■n ~ sn+1 ~~ ζη·
Then {Xn5 η > 1} is a strictly stationary sequence and satisfies any mixing
conditions mentioned in §1.1. The sum
£x* = £n+i-6 (II2)
has the characteristic function |/(£)|2 f°r aU n- Itls reasonable to introduce
some restrictions to make the variance of the sum Σ£=ι Хк increase when
η is increasing. Hence we assume always that Bn in (III) tends to infinity
asn-^oo.
First of all, we state the so-called Bernstein's blocking technique
utilized frequently in showing limit theorems for mixing random variables.
Let positive integers ρ = p(n), q = q(n) and к = k(n) with 1 < ρ < η, q =

40
Part II Weak Convergence
o(p), к — [η/(ρ + g)],1 and let
j>+(i-i)<7 j(p+q)
«=ϋ-ι)(ρ+ς)+ι «=ip+(i-i)q+i
η
%+ι = Σ Xi. (ИЗ)
i=fc(p+9)+l
Then
fc fc+l
Sn-Σο + Σ^· (II4)
By weak dependence, £1,62» ·'·> fife are asymptotically independent as g =
q{n) is large enough. On the other hand, the sum Σ^=\ Vj is negligible,
compared with Sn by noting q — o{p). Consequently, the Bernstein method
allows us to consider the sums of mixing random variables as independent
sums.
Using this method, by the procedure similar to that for an
independent sequence, for α-mixing sequence, we may prove the following theorem
about the class of possible limit distributions of sums.
Theorem III. Let {Xn, η > 1} be a strictly stationary α-mixing
sequence, {An} and {Bn} two sequences of real numbers with Bn —► 00 as
η —► oo. Suppose that the distribution function Fn(x) of the sum
±-J2xk-An
converges weakly to a distribution function F(x). Then F(x) is stable with
some exponent a. Moreover,
Bn = η1/α/ι(η),
where h(n) is a slowly varying function with positive integer argument.
*In this book, the sign [·] sometimes denotes the greatest integer part or at other
times denotes brackets. It will be clear from the context.

Chapter 3 Weak Convergence for
α-mixing Sequences
3.1 Necessary and sufficient conditions for the CLT
For an a-mixing sequence {Xn, τι > 1}, Ibragimov (1959, 1962) first
gave necessary and sufficient conditions for the central limit theorem (CLT).
In this chapter we always assume that {Xn, τι > 1} is a strictly stationary
a-mixing sequence unless special indication. Put Sn — Y^=1Xj, σ^ =
VarS„.
Theorem 3.1.1. Suppose that EX\ = 0 and EX\ < oo. Then in order
that {Xn} obeys the CLT and limn-^o σ^ — oo, it is necessary that
(1) σ\ = nh(n), where h(x) is a slowly varying function of the
continuous variable χ > 0,
(2) for any pair of sequences ρ — p(n), q = q(n) satisfying that as
η —> oo.
(a)p-+oo,q-+oo,q = o(p), ρ = o(n),
(b) n^Pq^Pp-2 -> 0 for all β > 0,
(c) np~1a(q) —► 0, and
hm —-
П-+00 ρσ£ J\x\>£(Tn
f x2dFp(x) = 0 (3.1.1)
for any ε > 0, where Fp(x) = P(SP < x). Conversely, if (1) holds and if
(3.1.1) is satisfied for some choice of the functions ρ and q satsifying the
given conditions, then the CLT is satisfied.
We do not prove this theorem here. For its proof we refer to Ibragimov
and Linnik's book (1971).
A simpler necessary and sufficient condition for the CLT was given by
Denker (1985).

42
Chapter 3 Weak Convergence for α-mixing Sequences
Theorem 3.1.2. Suppose that EX\ — 0, EX2 < oo and σ\ —► oo
as η —> oo. T/ien m order £/m£ {Xn} obeys the CLT, it is necessary and
sufficient that {S2/a„, η > 1} is integrable uniformly.
Proof. Necessity. Suppose that
5η/ση Д7У(0,1),
where iV(0,1) stands for a standard normal variable. Therefore, for any
ε > 0 there exists а К > 0 such that
lim / S2Ja2n dP= [ N2dP < ε,
n^°°J\Sn/an\>K J\N\>K
which implies uniform integrability of {S2/a^ η > 1}.
Sufficiency. Suppose that {S^/σ^, η > 1} is integrable uniformly. By
Theorem 2.1.3, we have
σ£ = ηΛ(η), (3.1.2)
where Л(ж) is a slowly varying function on [1, oo). Assume that ρ and
q are the functions satisfying conditions (a) and (b) in Theorem 3.1.1.
Furthermore, we choose ρ and q that satisfy condition (c) in Theorem
3.1.1 and
n2P~2<7q<7p 2 = n2p~3qh(q)/h(p) -► 0 as η -^ oo. (3.1.3)
For any given δ > 0, there exists a if > 0 such that
S2dP<6a2n (3.1.4)
J\s
l\Sn\>Kan
for each η > 1. By Property A4 of a slowly varying function,
σ1 ph(p) , .
-f = —— = o(l) asn -> oo,
σ~ ηη(η)
which implies that Κσρ < εση for all large n. Hence
Ρ J\Sp\>eGn У Ρ J\Sp\>Kap
< -δσΐ < 2δσ2
ρ Ρ η
The last inequality is due to (114), condition (c) and (3.1.3). Consequently,
condition (3.1.1) is satisfied. From Theorem 3.1.1, sufficiency is proved.

3.1 Necessary and sufficient conditions for the CLT
43
It is well known that boundedness of {E\Sn/an\2+6, η > 1} implies
uniform integrability of {S^/a2, η > 1}. The former requires existence of
higher order moments. This explains that the moment conditions imposed
on {Sn} are important for the CLT. A related result is given by Dehling,
Denker and Philipp (1986).
Theorem 3.1.3. Suppose that ΕΧχ — 0, EX\ — \^σ\ — nh{n), where
h(n) is a slowly varying function. Then in order that the distributions
°f {Sn/(7n, n > 1} tend to the standard normal distribution Φ(χ), it is
necessary and sufficient that
limsupan/£:|5n| < JV/2. (3.1.5)
n—>oo
In order to prove this theorem, we need the following lemma.
At first, we introduce some notations. Let integer ρ and real number
g satisfy
2<д<а~У\а^)АаУ\
where a(x) = α ([ж]), and let
ν2 = σ;2( S2dP, (3.1.6)
u2 = [ Si dP,
J\SP\<gTP
r = [g2d]
d satisfies 2g~x>2 V v2 < d < 1,
n = r(p + [σ*'4]) andr2 = ru2.
(3.1.7)
(3.1.8)
(3.1.9)
Lemma 3.1.1. Suppose that EX\ = 0 and EX2 = 1. We have
\Eexp(itSn/T) - exp(-f2/2)|
< 2d + \ί\σράι'2/η + |ί|3σρ/(ν/4) + |ί|3σ^/«3
+ 4α1/2(σ1/4)+ί4/5 + ί2σ2/Λ)
Proof. Note that и < σρ. Hence, for \t\ > r1/2, we have
|t|V(V/4) > r3/2/51/4 > ((92d - l)g-V°f2 > g2.

44
Chapter 3 Weak Convergence for α-mixing Sequences
Then the conclusion of Lemma 3.1.1 holds obviously.
Now we assume that \t\ < r1/2. Let q = [σν' ] and
jp+(i-i)g
0 = z2 Xi>
«=(i-i)(p+9)+i
з(р+я)
Recalling the definition of n, we can write
Sn = J2b+ &,='·& + %·
By the Minkowski inequality
ESf < r2a^ < 54σι/2 < σ3/2
Hence
\Eexp(itSn/r) - Eexp(itS'n/T)\
<|£exp(iiS£/r)-l|
< t2ESf/r2 < t2a3p/2/(u2r)
< t2o2p/{u2g).
Moreover, by Lemma 1.2.1, we have
\Eexp(itS'n/T) - (Eexp(itSp/r))r\
< 4ra(aV4) < 4α1/2(σ^4).
We now estimate \Eexp(itSp/r) — (1 — i2/(2r))|. By the Chebyshev
inequality we have
1 / exp(itSp/r) dP\ < g~2 < d/r.
U\Sp\>gap I
By Taylor's theorem
I / exp(itSp/r) dP-(l- t2/{2r))\
1 J\SP\<9<Tp '
< \P(\SP\ < gap) + - / SpdP
1 T ASp\<9<>p
( S2dP-(l-^)\
JlSJKaar, V V 2Γ/Ι
(3.1.10)
(3.1.11)
(3.1.12)
tA
2r2 J\SP\^P
|ί|8
>\Sp\<gop
+ l4[ \SpfdP
Τά J\SD\<oaO
(3.1.13)

3.1 Necessary and sufficient conditions for the CLT
45
Similarly to (3.1.12),
\l~P(\Sp\<gap)\<g-2<d/r.
Noting ESP = 0, we obtain
"IX
\SP\<9<r,
SpdP
SpdP
It is clear that
t2 f
'J\SP\>gap
< Cp/rg < apd1'2/(ru).
e
n , , s2vdP= n .
2t J\Sp\<gcP v 2r
The cubic term in (3.1.13) is estimated as follows.
■I
JaV
r ~ I \Sp\3dP
>91/2<rP<\Sp\<gap
< τ~39σ3ν2
< alv/(ru3)
and
/ \SpfdP
(3.1.14)
(3.1.15)
(3.1.16)
(3.1.17)
'|SP|<<71/2<rP
< r-tg^apu2
< apg^/iur3/2)
< ap/{rug1/4). (3.1.18)
Hence substituting (3.1.14)-(3.1.18) into (3.1.13) we obtain by (3.1.12)
\Eexp(itSp/r) - (1 - i2/(2r))| < η/r,
where
η = 2d + \t\apd^2/u + \t\3ap/(ug^4) + \t\3c3pv/u3.
Note that \ar - br\ < r\a - b\ for \a\ < 1, \b\ < 1. We obtain for |i| < r1/2
\Eexp(itSn'/r) - (1 - i2/(2r))r| < η + 4«1/2((Ji/4) (3.L19)
by (3.1.11). Moreover
|exp(-i2/2) - (1 - i2/(2r))r| < \ΐ\~ι for \t\ < r1'2,

46
Chapter 3 Weak Convergence for a-mixing Sequences
since \ex — (1 + x)\ < x2 for \x\ < \. Consequently, the lemma follows from
(3.1.19), (3.1.10).
Proof of Theorem 3.1.3. Necessity. If the distribution of Sn/an
tends to Φ (ж), then for any a > 0
/a \x\ 2/
-^=е~х l2 dx,
-а \/2тг
which implies (3.1.5).
Sufficiency. Let pn = y/n/2E\Sn\. If we can show that
Sn/Pn -^ JV(0,1) asn-^oo, (3.1.20)
then for any a > 0
/ N2dP= lim p~2 f S2ndP
J\N\<* n-+°° J\Sn\/Pn<*
<\imsupal/p2n < 1
n—>oo
by (3.1.5). Letting α —> oo yields that ση/ρη —> 1 as η —► oo. Hence
(3.1.20) implies 5η/ση Λ 7V(0,1).
Now we are going to prove (3.1.20). At first, for each η £ N we show
that there is an infinite sequence QcN and real numbers rn, η £ ζ) such
that
Sn/Tn -i 7V(0,1) η -> oo, η G Q. (3.1.21)
For this purpose we prove that there exist a sequence {g(p)> ρ > 1} and a
monotone sequence {c(p), ρ > 1} with the following properties:
#(ί0 "^ °°? c(p) ~~~> 0 as ρ —► oo,
υ2(ρ) := σ~2 / 52 dP -> 0 as ρ -► oo
y<,(p)V2<|Sp|/ffp<<,(p)
and
2ffb)"1/2 V v2(p) < c(p) < 1.
We first choose a sequence {z(p), ρ > 1} with
lim zip) = oo, z{p) < а~11\о1'А) A a\j*.
Next, we choose a sequence {i(p), ρ > 1} such that
i(p) -> oo, 2_i(p) log*(p) -^ oo asp -^ oo. (3.1.27)
(3J
(3.1
(3.1
(3.1
(3.1
L.22)
L.23)
L.24)
L.25)
L.26)

3.1 Necessary and sufficient conditions for the CLT
47
Fix ρ G N. Since the intervals U{p) := (z(p)2 % , z(p)2 *], 0 < i < i(p),
are disjoint, there exists an integer к = k(p) with 0 < к < i(p) such that
σ;2 ί S2pdP<l/i{p). (3.1.28)
p J\sp\/<rpeik{p) p
Let
g(p) = z{pf-4p). (3.1.29)
Then g(p) -*■ oo as ρ -► oo by (3.1.27). Because of (3.1.26)-(3.1.28),
(3.1.23) and (3.1.24) axe satisfied. Since 25(p)-1/2 V v2(p) -*■ 0 as ρ —> oo
we can choose {c(p)} with c(p) | 0 and satisfying (3.1.25).
With these choices of {g{p)} and {c(p)}, we define u(p), r(p) and n(p)
by (3.1.7), (3.1.8) and (3.1.9) respectively. Put Q = {n(p), ρ > 1} and
define 7-2, η G Q by (3.1.9). Since
σ;1^! = σ~ι [ \SP\ dP + σ~ι [ \SP\ dP
J\Sp\/ap<g(p) P J\SP\/ap>g(p)
< a^u(p) + g(p)-\ (3.1.30)
we have, by (3.1.5), for ρ large enough
η{ρ)/σρ > 7/2, (3.1.31)
where 7 = inf{E\Sp\/ap, ρ > 1} > 0. Lemma 3.1.1 now implies (3.1.21).
Next we show that
lim τη/ρη = 1. (3.1.32)
To see this we choose a sequence {b(m), m > 1} with
lim b(ra) = 00,
m—>oo
lim 8ир{|Л(«т)/Л(т) - 1|, 1 < ί < 6(m)} = 0. (3.1.33)
This is possible. Indeed, by the Karamata theorem (see Appendix Theorem
Al) there exists an increasing sequence {га&, к > 2} such that
sup h(tm)/h(m) — 1
i<t<k
< -, m > mfc.
Then £>(·) defined by Ь(т) = к for m^ < m < m^+i has the desired
properties. Of course, we can assume that {z(p), ρ > 1} is chosen so that

48
Chapter 3 Weak Convergence for α-mixing Sequences
in addition to (3.1.26) we have z{p) < b(p)1/2/2. Then for all large ρ we
have, by (3.1.9) and σ2<ρ2,
σ2(η(ρ)) = r(p)(p+[al/4])h(r(p)(p+[al/4}))
r(p)a2 r{p)ph{p)
= (1 + 0(р-1/а))Мг(р)(р+[аУ4])) = χ +
h(p)
by (3.1.33). Thus, by (3.1.9) and (3.1.30) we have for ρ large enough
E{Sn(p)/rn{p))2 = σ2η(ρ)/τ2η{ρ) < 2a2/u(p)2 < 8/72. (3.1.34)
Hence {Sn/rn, η € Q} is uniformly integrable and thus, by (3.1.21),
lim E\Sn\/rn = E\N\ = J2/^.
n->oo,nGQ v
This proves (3.1.32).
If Q = N, (3.1.21) and (3.1.32) imply
Sn/pn -^ N(0,1) as η -> oo. (3.1.35)
Consider the case of Q С N. Assume that {z(p), ρ > 1} constructed in
the proof of (3.1.21) satisfies
z(p) < wm{a{all*)-V™, σ^16, ρ1/4, b(p)^/2) (3.1.36)
and
z(p) < z(q) < z{pfl\ p<q<p2. (3.1.37)
Such a sequence can be constructed as follows: First choose an increasing
sequence y{p) satisfying (3.1.36). By induction on к define z(p) = y(pk) Л
z(pk-if2 for ρ = рк = 22\ pk + 1,. · .,pfc+1 - 1. Let {/(η), η £ Q}
and {j(n), η £ Q} be two arbitrary sequences of real numbers tending to
infinity and with l(n) < j(n), η £ Q.
Recalling condition (3.1.5) and noting JS|5n| < ση, we have
лД/тг < limsupan//9n < 1. (3.1.38)
n—>oo
Moreover, we have shown that Sn/pn —> -?V(0,1), η —> oo, η £ Q·
Consequently, we obtain, for any a > 0
/ N2dP= lim [ S2JpldP
J\N\<oc neQJ\Sn\/pn<a
<liminfp-2/ 52dP,
"SQ J\S„\/an<l(n)

3.1 Necessary and sufficient conditions for the CLT
49
which implies by letting a —► oo
l<liminf/9-2 / S2ndP
neQ J\Sn\^n<l(n)
<limsuP/9-2/ SldP<\. (3.1.39)
neQ J\Sn\/a„<j(n)
(3.1.38) and (3.1.39) imply
lima"2 / SldP = 0. (3.1.40)
«eQ Jl(n)<\Sn\lan<j(n)
We shall apply Lemma 3.1.1 again. To prepare for it we set
l{n) = l(n(p)) = z(n(p)) if η = n(p), ρ > 1 (3.1.41)
and
j(n) = j(n(p)) = min(a-1/4(orV4)) <Ti/4> Ъ{п)^/2)
if η = n(p), ρ > 1. (3.1.42)
For ρ large enough, we have by (3.1.22), (3.1.29) and (3.1.36)
n{p) < 92{p)c{p)(p + p1/4) < z2(p)p < p1/2p < p2. (3.1.43)
By (3.1.36), (3.1.42) and (3.1.41)
j(n) > z\n) = l\n) > l(n), n€Q. (3.1.44)
Since, by (3.1.41)
w2
(n):=a~2 [ S2ndP-^0, η € Q, (3.1.45)
we can choose a nonincreasing sequence {d(n), η G Q} such that
lim d(n) = 0 and d(n) > 2Λ(η)_1/2 Λ w2(n), neQ. (3.1.46)
neQ
Let Q = {nfc, к > 1} be arranged in increasing order and let Jk be the
interval
Jk = [nkl2(nk)d(nk), nkj2(nk)d(nk)].
We show that there exists а ко such that
Jk П Jfc+i φ 0, к > к0. (3.1.47)

50
Chapter 3 Weak Convergence for α-mixing Sequences
Obviously n{nk) = г{пк){пк + [^(^fc)1^4]) € Q· As n^+i is the smallest
member of Q bigger than nk we must have for large к
пк+1 < п(пк) < nkz2{nk) < n\
by (3.1.43). Hence, by (3.1.41) and (3.1.37), the left endpoint of Jk+1 does
not exceed
nk+1l2(nk+1)d(nk+1) < nkz2(nk)z2(nk+1)
< nkz2(nk)z2(n2k) < nkz5(nk)
for large k. On the other hand, by (3.1.46) and (3.1.44), the right endpoint
of Jk is bigger than
nkj2(nk)d(nk) > nkj2(nk)r1/2(nk) > nkz(nk)15/2
for large k. Since z(nk) -^oowe obtain (3.1.47). Let m > min{/, / G Jko}·
Then there is а к > ко such that m G Jk. Thus we have for some g G
\Knk), j(nk)] and some |0| < 2
m = g2d(nk)nk = [g2d(nk)](nk + [a1/4(nfc)D + впк
=:Мк+впк. (3.1.48)
Now by (3.1.8), Mk is of the form (3.1.9) and hence we can apply Lemma
3.1.1. Put ρ = nk and d = d(nk). By (3.1.31), u(nk)/a(nk) > \η > 0.
Now g > l(nk) —> oo and a(altA{nk)) —> 0. Finally, by (3.1.45) and noting
l(nk) < g < j(nk), we have
v2(nk) := a"2(nfc) / 5^ dP < w2(nfc)
0.
V/2<|5nJ/^(nfc)<^
Hence, by Lemma 3.1.1,
SMh/T(Mk)±N(p,l). (3.1.49)
Since |0| < 2 we have by (3.1.33) for large к
ES\6\nk < \θ\ηφ(\θ\η,) ^ Α
a2(nk) ~ nkh(nk)
Consequently
\2 ζτό2 ^ л ~2
E(Sm - SMky = ES(e]nk < Aa\nk). (3.1.50)

3.2 Sufficient conditions for the CLT and WIP
51
Denoting r*(nk) = [g2d(nk)] we obtain by (3.1.33) for large к
а2(Мк) > r*(nk)nkh(r*(nk)(nk + [σ1/4^)])) > 1
r*(nk)a2(nk) ~ r*(nk)nkh(nk) ~ 2
since r*(nk) < g2d(nk) < j2(nk)d(nk) < b(nk)/2 by (3.1.41 ). Hence, from
(3.1.50) and as r*(nk) —> oo, we have
E(Sm - SMk)2/a2{Mk) -^ 0 as к -> oo. (3.1.51)
In the same way as (3.1.34) one can prove
a2(Mk)/r2(Mk) < 8/72. (3.1.52)
Put
т~т = T(Mk), if?77,andMfc are as in (3.1.47).
Then by (3.1.49), (3.1.51) and (3.1.52)
Srn/rrn -^ Λ^(0,1) as πι -> oo. (3.1.53)
The sequence {Sm/Tm, m > 1} is uniformly integerable in view of (3.1.51)
and (3.1.52). We obtain (3.1.35) from (3.1.53) and (3.1.32).
With the help of it and similarly to (3.1.39) we have ση/ρη -> 1 as
η —► oo. This completes the proof of Theorem 3.1.3.
3.2 Sufficient conditions for the CLT and WIP
In the last section, we give some necessary and sufficient conditions
for the CLT. But it is not easy to verify them. In this section, we shall
give some sufficient conditions for the CLT and weak invariance principle
(WIP). Rosenblatt (1956) first gave sufficient conditions for the CLT. After
that many authors (e.g. Ibragimov 1962) have discussed this subject and
have obtained some further results. One of the best is due to Herrndorf
(1984, 1985). The following theorem is attributed to Gordin (1969) and
has been restated and proved by Hall and Heyde (1980) (Corollary 5.3(ii))
via approximating Sn by a naturally related martingale with stationary
ergodic differences. Its proof will not been presented here.
Theorem 3.2.1. Suppose that EXX = 0, Е\Хг\2+6 < oo for some
δ > 0 and
oo
Σ <*(п)6/(2+г) < oo. (3.2.1)
n=l

52
Chapter 3 Weak Convergence for α-mixing Sequences
Then
oo
σ2 := EX\ + 2 ^ £ΧχΧ,- < oo (3.2.2)
i=2
and, i/σ^Ο,
5п/ал/^Д ΛΓ(Ο,Ι). (3.2.3)
Remark 3.2.1. For the case of bounded variables, i.e. 5 = oo,
condition (3.2.1) is reduced to
oo
Σ a(n) < °°· (3.2.4)
n=l
Remark 3.2.2. Davydov (1973) gave two examples which pointed
out that the rate of a(n) in Theorem 3.2.1 could not be improved in certain
senses.
Example 3.2.1. For any δ > 0 and ε > 0, there is a strictly
stationary countable-state Markov chain {Xn, η > 1} with EX\ — 0, £|Xi|2+<5 <
oo such that
(i) a(n) = oin-^-^+W) as η -^ oo,
(ii) VarSn « nd ^ for some 1 < d < 2,
(iii) Sn is attracted to a symmetric stable law with exponent a, 1 <
a < 2.
Example 3.2.2. For any ε > 0 there exists a strictly stationary
countable-state Markov chain {Χη? η > 1} with EX\ — 0, |Χχ| < cq a.s.
for some Co < oo such that a(n) = ο(η~(1-ε^) as η —> oo and properties
(ii) and (iii) in Example 3.2.1 hold.
We now investigate the weak invariance principles. Define random
elements on D[0,1] as follows:
Wn(t) = S[nt]/an, 0<ί<1.
Convergence theory of probability measures tells us that the key to
the proof for weak invariance principle lies in verification of tightness. One
of the following conditions is sufficient for tightness (cf. Billingsley 1968,
Section 16).
(1) For any ε > 0, η > 0, there exist a 5, 0 < δ < 1, and an integer no
such that, for 0 < t < 1
\p{ sup \Wn{s)-Wn{t)\>e)<^ n>n0. (3.2.5)
О lt<s<t+6 }
λα w b means lima/6 = 1.

3.2 Sufficient conditions for the CLT and WIP
53
(2) For any ε > 0, there exists a λ > 1 and an integer no such that
PJmax \Si\ > \ση\ < ε/λ2, n>n0. (3.2.6)
Davydov (1968) first generalized the CLT to the invariance principle for
bounded variables, but condition (3.2.4) was strengthened as Σ™=ι α1/2(η)
< oo. Oodaira and Yoshihara (1972) sharpened his result and obtained the
invariance principle under the conditions for the CLT basically.
Theorem 3.2.2. Suppose that EXX = 0, \Хг\ < c0 < oo. J/ (3.2.4) is
satisfied and a{n) < c/nlogn. Then, if σ > 0,
i.e. Wn weakly converges to a Wiener process W with ση = ал/п.
Proof. By Remark 3.2.1, it follows that Wn(t) converges to W(t) in
distribution for any £, 0 < t < 1. By the Cramer-Wold method, it is easy
to see that for any given 0 < t\ < £2 < * * * < tk ^ 1> (^n(^i)5 * * *> Wn(tk))
converges to (W(£i),· · ·, W{tk)) in distribution.
We now show the tightness of {И^}. It is enough by (3.2.6) to prove
that
P{ max \Si\ > SXay/n] < ε/λ2, η > n0. (3.2.7)
Put ρ = [vV(logn)3/8], к = [η Ι ρ]. We have
Р{\Хг\ + · · · + \X2p\ > λσ^} = 0 (3.2.8)
for large η by boundedness of {Xn}· Moreover, using Lemma 1.2.1 and
condition (3.2.4) we have uniform integrability of {52/n, η > 1}. Therefore,
for any ε > 0, there exists a λ > 1 such that for each г > 1
P{\Si\ > \σ\Γί) < ε/3λ2. (3.2.9)

54 Chapter 3 Weak Convergence for α-mixing Sequences
Put Ej — {maxi<i<j \S{\ < SXa^/n < \Sj\}. We have
P< max \Si\ > 3Xay/n\
l 1<г<п J
η
< P{\Sn\ > λσ^} + p((J {Ej f](\Sn -Sj\> 2λσ^)})
3=1
< P{\Sn\ > \ay/n)
k-2 ρ
+ Σ P(U {Яр+з П(15- " s*p+i\ ^ 2λσ^)})
i=0 j=l
η
+ £ Ρ{|5η-5^·|>2λσν^}
j=(fc-l)p+l
< Ρ{|5„| > λσ^η}
+ Σ Ρ{U {^ Π(Ι5« - %Η2)„Ι > λσ^)})
г=0 j=l
Ρ
η
+ £ ^{|*ι| + ··· + |*η-,·|>2λσν^}
j=(fc-l)p+l
< Ρ{|5„| > λσν^}
+ ΣΡ{(ύ ^И-i) Π(ΐ5« - 5(i+2)Pl > λσν^)}
i=0 j=l
+ 2ηΡ{\Χχ\ + ■■■ + \Χ2ρ\ > λσ^}
=:/ι + /2 + /3. (3.2.10)
By (3.2.8), Ιζ = 0. (3.2.9) implies that /χ < ε/3λ2. Since U?=1Eip+j G
^i+1)p, (|5n - 5(i+2)p| > \σφϊ) € ^+2)p+1, we obtain
k-2 ρ
72 ^ Σ P{ U ^ii>+4P{l5" - 50+2)Pl > λσ^} + ka{p)
i—o j=l
< ε/3λ2 + ka(p)
by (3.2.9) again. Prom a(n) < c/n log n, it follows that
en
n(logn)~'3/4log(n1/2(logn)~3/8)

3.2 Sufficient conditions for the CLT and WIP
55
Inserting these estimations into (3.2.10) yields (3.2.7). The proof of
Theorem 3.2.2 is completed.
Some authors have discussed and extended this theorem. The
general result was given by Herrndorf (1985). He removed the assumption of
stationarity and made the moment condition more flexible. Denote
Q = {g{x) : [0, oo) —► [0, oo), g{x) is convex;^(0) = 0,
g(x)/x2is non-decreasing, lim g(x)/x2 = oo}.
χ—»οο
For every g £ Q we define the inverse inv# : (0, oo) —> (0, oo) by g(mv g(x))
= χ and fg : [0, oo) —> [0, oo) by fg(0) = 0 and
fg(x) = (inv (7(1/2;)) χ for χ > 0.
Theorem 3.2.3. Let {ХП5 η > 1} be an α-mixing sequence with EXn
= 0, EX2 < 00 for alln>\ and
ESl/n-^σ2 as η -^ oo (3.2.11)
for some σ > 0. If there exists a g £ G such that
00
sup^(|Xn|) < 00, Σ fg(a(n)) < °°> (3.2.12)
^ 71=1
then Wn => W.
The proof of Theorem 3.2.3 needs the following lemmas.
Lemma 3.2.1. Let £1, · · ·,£η be random variables. Put
a = max sup{\P(AB) - P(A)P(B)\ : A G σ(&, · · ·,&),
l<fc<n—1
ΒΞσ(&+ι,...,£η)}.
Γ/ien /or any ε > 0,
Σ
i=i
pfeJ^I>24
< Ρ{|Σ?=1^Ι>ε} + ηα

56 Chapter 3 Weak Convergence for α-mixing Sequences
Proof. Put
Ax = {|6|>2e},
к ι
Ak = {|Σθ| >2ε' |Σθ| <2e, 1 <ί < fe-l}, 1<к<п.
η η
β* = {| Σ &|<e}' !<*<^ 5η = «, £ = {|Σ&|>ε}·
i=fc+i i=i
It is easy to see that \У^=1А^В^ С С and
\Р{АкВк) - P(Ak)P(Bk)\ < a.
Hence
P(C) > Σ P(AkBk) > min P(Bk) Σ P(Ak) - na. (3.2.14)
Note that
fc=i ^*** *=г
Combining it with (3.2.14) yields (3.2.13).
Lemma 3.2.2. Let {Xn, η > 1} be as in Theorem 3.2.3, and satisfy
sup E(Sm+n - Srn)2/n < oo. (3.2.15)
n>l,m>0
Assume that there exist positive integers ρ = p(n), q — q(n) such that as
η —► oo
ρ = o(n), g = o(p), np^aiq) = o(l), (3.2.16)
η-1 ^ |£Χ;Χ,·|-^0, · (3.2.17)
i<i,j<n,\i-j\>q
p~x max £{(Sm+p-Sm)2J(|Sm+p-Sm| > εη1/2)} —► 0 for any ε > 0.
0<m<n—p
(3.2.18)
ГЛеп <Λβ CLT holds.
If moreover, for any ε > 0
np * max P<{ max |5m+r — Sm\ > ε^/η\ —> 0 (3.2.19)
0<m<n—ρ ^1<г<ю J
#ien Wn => W with ση = σ^/η.

3.2 Sufficient conditions for the CLT and WIP
57
Proof. Denote к = [η/(ρ + q)]. к —> oo as η —> oo by (3.2.16). Put
j(p+q)+p
i=j(p+q)+l
U+l)(P+9)
4j = Σ X*' 0 < j < A: - 1,
η
i=k(p+q)+l
In order to prove the CLT, it suffices to show that as η —> oo
(a)£S»2/n-0,
(b)
(c) pexp(iiS^ - Γΐ)=ο Eexp(it^j)\ —► О uniformly in t G (-00, 00),
(d) Σ?=ο £($/(!£, Ι > εσηχ/2))/η _> 0 for any e > 0.
Prom Lemma 1.2.3 and (3.2.15) we have
ESf<J2Ev] + 2J2 Σ \EXiXi\
j=0 г=1 i+p<j<n
<c(kq + p + q) + 20n ^ /^(а(г)) sup ||X,||
г=р+1
2
Now (a) follows from (3.2.12) and (3.2.16). By the same way, we obtain
(b). As to (c), we have by Lemma 1.2.1 and (3.2.16)
fc-l
Eexp(itS'n) - Yl JSexp(iifj) < 16ka(q) -> 0 as η —► oo.
Finally, (d) follows from (3.2.18) immediately. Thus the CLT holds true
for {Xn}.
Let 0 < t\ < · · · < tk < 1 be given. We wish to show
(^п(^),.-.,^п(^))-(^1),·--,^^)) as n-oo (3.2.20)
in distribution. Prom (3.2.11) and the CLT, Wn(t) converges to W(t)
in distribution for any £, 0 < t < 1. Therefore {Wn(£i),· · -Дп(^)} is
tight by Prohorov's classical characterization of tightness. Let Q be the

58
Chapter 3 Weak Convergence for α-mixing Sequences
limit distribution of some subsequence of {Wn(£i), · · ·, Wn(£fc)}, and nti
the mapping that carries the point χ = (χχ, · · · , ж*.) of Rk to the point Xt{
of R. According to the CLT, the marginal distribution Qtt^1 are normal
with variance t{. Take rn = q/n, then rn —> 0 and an(q) —> 0, where
an(g) = sup{|P(AS) - Р(Л)Р(В)| :
A G σ(Χ;, 1 < г < га),
Ρ G cr(Xi, m + q <i <n, 1 < m < η — q}.
Using (3.2.15) one obtains £(Wn(^ + rn) - Wn(U))2 -► 0 as η —► oo.
Hence Q(7rtl, πί2 — ntl, · · ·, 7rifc — πίΑ._1)~ is the limit distribution of some
subsequence of (Wn(«i), Wn(t2) - Wn(«i + rn), · · ·, W(tfc) - Wn(tfc_i +rn)).
Now an([nrn] - 1) —► 0 implies that π4ι, πί2 - πίΐ5· · ·,πίΑ. - щк__г are
independent under Q. Therefore Q is the distribution of (W(£i), ···, W(tfc))·
This argument proves (3.2.20).
Finally, we have to prove the tightness of the sequence {Wn}. It suffices
to show a version of (3.2.5), i.e., for any ε > 0, η > 0 there exist a 5, 0 <
δ < 1, and an integer no such that
[1/δ] r ι
Σ Mr ,,i ^x,, , J5' - 5н^]1> εσ^ί < *> n^ n°- (3·2·21)
r^l ^[пк6]<г<[п(к+1)6] >
Let A; G {0, · · ·, [1/5]} be fixed, m = m(n) = [([n(k + 1)δ] - [nkS])/(p + q)],
[nk6]+j(p+q)+p [nk6] + (j+l)(p+q)
£j= Σ XuVj= Σ Xi> J = 0,l,---,m-1.
i=[nk6]+j(p+q)+l i=[nk6]+j(p+q)+p+l
Then we have
ΡI max \Sr — S\nkfi]\ > eay/n\
<(m + l) max P<( max |S;+r — SA > εσ^/η/3>
~V 0<l<n-(p+q) U<r<p+q] ^ ' ' J
r
+ P( max lyfJ > еау/п/з]
3=0
r
+ P{ max У^Лд\> еау/п/ЭЛ
lo<r<m-il^ -Ί ' J
=: /i + /2 + /3. (3.2.22)
From (3.2.19), it follows that asn->oo
1\ < cnp~l max P{ max |S/+r — 5/1 > еал/^/з[ —> 0.
~ 0<i<n-(p+g) ll<r<p+g' '+Г £| V ' J

3.2 Sufficient conditions for the CLT and WIP 59
By Lemma 1.2.3, (3.2.15) and (3.2.12), we obtain
^аХт~г}Е^У/{(х2п)
< £ Εη2/(σ2η) + 2 £ £ \ЕХ^\/{а2п)
0<j<m—l l<i<ni+p<j<n
< cmq/(a2n) + 20σ~2 £ fg(a(j)) sup \\Xj\\2g -► 0.
Similarly
max Ε(ΥξΛ2/(σ2η)
JC{0,.,m-l} V^4V
< £ £^/(σ2η) + 2 χ) ^ \ЕХ^\/(а2п)
0<j<m—l l<i<ni+q<j<n
< cmp/(a2n) + 20σ~2 £ /ff(a(j)) sup ||Χ,·||2.
Using the definition of m and (3.2.12), we obtain that the last expression
converges to ca~26 as η —> oo. Choose 6ο(ε) > 0 such that
(18/ε)2οσ-%(ε) < \-
Prom now on we shall assume δ < δο(ε). By the Chebyshev inequality we
have
m—1 1
ο<ϊη<^-2ρ{|ΣιΦ-^/6}>-
- - J=r+1
for large n. Now we apply Lemma 3.2.1 and obtain
m—1
/2 < 2P{| Σ ξ,·| > εσν/η/б} + 2man(? + 1).
i=o
(3.2.16) implies man(g + 1) —► 0 as η —► схэ. Since
m—1
i=o
m—1 „
E[s[n{k+1)s] - S[nkS] - Σ (£i + %)) /(σ2η) -» ° asn -^ oo,
i=o

60
Chapter 3 Weak Convergence for α-mixing Sequences
we obtain
r
lim sup P| max \S^CA > εσ\Ζΰ/3\
η_+οο^ lo<r<m-i\^J\ v ' )
3=0
< 21imsupP{|s[n(fc+1)(5] - S[nk6]\ > еау/п/т\.
Applying the Chebyshev inequality and Lamma 3.2.1 again yields
r
lim sup P{ max Υ"" m > εσ^/η/Ζ \
n-юо l0<r<m-llr-^ I >
.7=0
m— 1
< 2 Km sup Ρ\ Y^ щ > eay/n/6 > + 2 lim sup man(p + 1) = 0.
n—>oo · л η—юо
Summing up these results for к = 0, · · ·, [1/6], we obtain
lim sup У^ P< max 5r — S\nkn\ > εσ-ν/η?
«-«> έο 4n*«]<r<[n(fc+i)«]l l Ί J
< 2 V lim sup P{ S[n(fc+i)i] - 5[nfc6] > eay/n/7\
fc=o n^°° u ' J
<2(^ + l)P{|JV(0,<5)|>e/7}-^0 as<5-^0.
0
Thus Lemma 3.2.2 is proved.
Proof of Theorem 3.2.3.
Assume that some g £ G satisfies condition (3.2.12). Then К :=
SUPn>l \\Xn\\g < OO.
We note that the conditions (3.2.18) and (3.2.19) of Lemma 3.2.2 are
both implied by
m+p m+p
Ρ_1ο<^-/((Σ l*l)'( Σ \Хг\>е^))^0,
i=m+l i=m+l
as η —> oo. (3.2.23)
From the monotonicity of g(x)/x2, the convexity of # and ^(|Х|/||Х||^) <
1 if 0 < \\X\\g < oo, we have
m+p m+p
Ε((Σ \X,\) l( Σ \Xi\>eV^))
i=m+l i=m+l
m+p
< Eg{ Σ \Xi\/Kp)e2n/g(e^/KP)
i=m+l
< e2n/g(ey/n/Kp).

3.2 Sufficient conditions for the CLT and WIP
61
Thus (3.2.23) is implied by
p-1n/g{ey/^IKp) -> 0 as η -> oo. (3.2.24)
From Lemma 1.2.3 we have for m > 0, η > 1
oo
E(Sm+n - Sm)2 < nsnpEX2 + 2nK2 Σ /*(<*(*))·
Since supJ>x £X? < oo, we obtain (3.2.15). The proof of Theorem 3.2.3
will be completed by constructing sequences p(n), q(n) such that (3.2.16),
(3.2.17) and (3.2.24) are fulfilled. Since к —► fg(a(k)) is non-increasing,
the assumption Y%Li fg(a(k)) < oo implies fg(a(k))k —► 0 as к —► oo.
Therefore we can choose a : [0, oo) —► [0, oo) continuous and strictly
decreasing such that
fg(a(x))x-^>0 as#—>oo, (3.2.25)
a(x) > a(k) for all integers к > χ. (3.2.26)
Since χ —> xmv g(\/a(x)) homeomorphically maps (0, oo) on (0, oo), we
can define x(n) G (0, oo) by
x{n) inv g(l/a(x(n))) = nly\
Then x(n) —> oo as η —> oo. (3.2.25) implies that there exists an L =
L(n) > n-1/4 with L(n) —> 0 as η —> oo such that
L(n)~2 sup (ίην#(1/α(*)))2α(*)£ -> 0 as η -> oo. (3.2.27)
£>cc(n)
Define у = y(n) G (0, oo) by
y(n)invg(l/a(y(n))) = L(n)n1/2.
Clearly y(n) > x(n) and y(n) = ^(n1/2). Put q{n) = min{j, j > y(n)} and
choose a sequence p(n) such that asn->oo
p(n)/y(n) -+ oo, p(n)/n -+ 0, L(n)p(n)/y(n) -+ 0. (3.2.28)
Now ρ = p(ri) = o(n), q = q(n) = o(p) hold true. Since q(n) —> oo, (3.2.17)
can be obtained from the assumption of the theorem by an application of
Lemma 1.2.3. Using (3.2.26), the definition of y(n) and (3.2.27) we obtain
nq~1a(q) < ny~1a(q) < ny~1a(y)
= L~2(invg(l/a(y)))2a(y)y -► 0 asn->oo, (3.2.29)

62
Chapter 3 Weak Convergence for α-mixing Sequences
which and q = o(p) imply np~1a(q) = o(l). For all large η we have ρ > у
and ye/KpL > 1 by (3.2.28). For such η we obtain by the convexity of g
and the definition of y(n) :
p^n/giey/n/Kp) < y^npLK/ieygiLy/n/y))
= y~1na{y)pLK/(ey).
This expression tends to 0 by (3.2.28) and (3.2.29). Hence (3.2.24) holds.
The proof of Theorem 3.2.3 is completed.
The following corollaries are immediate. Let g(x) = χ2+δ for some
δ > 0 in Theorem 3.2.3.
Corollary 3.2.1. Let {ХП5 η > 1} be an α-mixing sequence with
EXn = 0,
oo
sup E\Xn\2+s < oo, 53 <x(n)6/{2+6) < oo for some δ > 0.
n=l
Suppose that condition (3.2.11) is satisfied. Then
Wn^W.
Corollary 3.2.2. Let {Xn, n> 1} be an α-mixing sequence with EXn
= 0.If
OO
supEX2\\og\Xn\\a < oo, 53|loga(n)ra < oo (3.2.30)
n=l
/or some a > 0, particularly
supEX2\\og\Xn\\a < oo, a(n) = 0(6_n) (3.2.31)
П
/or some a > 1 and Ь > 1, Йеп
Herrndorf (1985) has given two examples, which show that a > 1
cannot be replaced by a = 1 in (3.2.31). The constructions of the examples
are omitted here.

3.3 The CLT and WIP when the variance is infinite
63
Remark 3.2.3. Doukhan, Massart and Rio (1994) discussed the
functional CLT for α-mixing sequence, via the α-mixing function a(t) and
the tail distribution function of |Χχ|, they gave another sufficient condition
as follows:
Let {Xn,n > 1} be a strictly stationary α-mixing sequence of centered
random variables satisfying
/ mva(u)[invG(u)]2du < oo (3.2.32)
Jo
where G{u) = P(\Xi\ > u). Then the series σ2 = Σ£=1 Cov(XbXn) is
absolutely convergent, and
Ζη/σ =► W,
where Zn(t) = ^ ЕЙ Хк-
Particularly, if \Χχ\ < с < oo, condition (3.2.32) is equivalent to (3.2.4).
So, in this case, Theorem 3.2.2 is recovered.
3.3 The CLT and WIP when the variance is infinite
The CLT for a mixing sequence with possibly infinite variance was first
discussed by Lin (1981, 1982). He proved the following theorem under the
conditions comparable with that in the case of independent sums. Assume
also that {Xn5 η > 1} is a strictly stationary α-mixing sequence.
Theorem 3.3.1. Suppose that EX\ = 0 and the following conditions
are satisfied:
(i) There exist two sequences of positive integers ρ = p{n) andq = q(n)
satisfying
ρ = o(n), q = o(p), ka(q) —► 0 asn-> oo, (3.3.1)
where к = k(n) = [n/(p + q)].
(ii) There exists a sequence of constants {Cn} with Cn | oo and
к [ dP = o(-), (3.3.2)
J\Xi\>Ck ^P'
οί/("Ιχ^χ?άρ)=°Φ· (3·3·3)

64
Chapter 3 Weak Convergence for α-mixing Sequences
uniformly in i. Then, {Xn} obeys the CLT.
If moreover there exists a constant a > 0 such that ρ < ka and
lim k2+aa(q) = 0, (3.3.5)
η—>οο
Йеп Йеге are constants Bn > 0 5?xc/i £/m£ {Wn(t) = £[ηίι/Βη, 0 < £ <
l}n>i weakly converges to W.
Proof. Define £J5 j = 0,1,··, A; — 1, and r/j, j = 0,1,· · ·,& as in
Lemma 3.2.2. From (3.3.2) and (3.3.3) we obtain
Hence
Cl [ dp/ [ X\ dP = o(p-2). (3.3.6)
J\Xi\>Ck I J\Xx\<Ck
0</ &dP-( ildP
%о|<рС*)П(и*=1(|Х,|>С*))
< ръС\ [ dP = o(p f X\ dP),
J\Xi\>Ck V ./|Xi|<C» У
i.e.,
/ $dP= [ (ΈΧ1 + 2 Σ MiW
+ ofp/ XldP). (3.3.7)
V ^|ATi|<Ct У
Furthermore, from (3.3.6) again
0 < / X? dP - / A"? dP
= / X?dP
А|^1<с*)пи^«р(|х4|>сь)
<pC|/ dP = o(p-1 [ XfdP).
Therefore
ν
Σ / *;dP
^ί^η?=1(|Λ<|<σ»)
= p f XfdP + o( [ Xf dP). (3.3.8)
J\Xi\<Ck \J\XdKCk '

3.3 The CLT and WIP when the variance is infinite
65
Similarly
/ XiXj dP
JrPj=i(\Xi\<Ck)
= ί Χ,Χ^άΡ + οίρ-1 [ XfdP).
./|X<|<Cfc,|Xf|<CJb V J\Xi\<Ck '
Using condition (3.3.4), we obtain
У [ XiXidP = o(p( XfdP). (3.3.9)
l<i<j<pJpli=iUXj\<Cb) J\Xi\<Ck '
Now (3.3.7), (3.3.8) and (3.3.9) imply
/ $ dP = (1 + o(l))p / Xf dP. (3.3.10)
J\to\<pCk J\Xi\<ck
Moreover,
/ dP< [ dP<p [ dP. (3.3.11)
Ato\>pCk ^=1{\Ха\>Ск) J\Xi\>Ck
Then using conditions (3.3.3) and (3.3.2) we obtain
т-^Цз/ ξ%άΡ^οο, (3.3.12)
к [ dP-^0. (3.3.13)
Аы>рск
According to (3.3.10), L· \<pC £o dP —► oo as η —> oo. Imitating the proof
of Theorem 35.1 in Gnedenko and Kolmogorov (1954), we have
(/ ξ0άΡΫ = ο([ f0dP),
yJ\to I <pCk ' yJ^o\<pCk '
i.e., (3.3.12) is equivalent to
Τ^γΛ I iUP-U todP)2} - oo. (3.3.14)
Let £·, j = 0,1, · · ·,& — 1, be independent random variables with
the same distribution as £o· By checking the proof of Theorem 26.4 in
Gnedenko and Kolmogorov (1954), under conditions (3.3.13) and (3.3.14),
(p)
there exists a sequence of positive constants B^ such that
^^4jV(0, 1) asn-^oo. (3.3.15)
o(p)

66 Chapter 3 Weak Convergence for α-mixing Sequences
In fact, we can take
B^ = k{[ &dP-([ ξοάΡ)2}
4
= (l + o(l))fc/ $dP
= (1 + o(l))kp [ Xl dP (3.3.16)
J\Xi\<Ck
by (3.3.10). Using Lemma 1.2.1 and (3.3.1) we have
fc-i fc-i
\Eexp(it Σ ti/B^) ~ Π EexpWj/B^l < 4ka(q) - 0. (3.3.17)
j=o j=x
Combining it with (3.3.15) yields
, fc-i
-щΣοдN^!) ™n- ~· (3·3·18)
Bk j=0
Repeat the above discussion for 7^·, j = 0,1, · · ·, к — 1. Then there exists a
в[я) > 0 such that
, k-1
-7т^ч;4 ЛГ(0, 1) as η -> oo. (3.3.19)
β
к i=o
Similarly to (3.3.16), S^ = (1 + o(l))ifeg/|Xl|<Cjb X?dP. Therefore
(ВР/ВР)2 = (1 + o(l))g/p = o(l). (3.3.20)
Thus
fc-i
^ Σ *&' ~* ° as η -^ oo. (3.3.21)
Bk ' j=o
Moreover, noting η — k(p + q) < ρ + q and using (3.3.16) and (3.3.3) we
have for any ε > 0
Р{\щ/В^\> ε} <(р + д)Е\Х1\/еВР
< 2рЕ\Х1\/е(кр [ XfdPY/2 = оШСк) -> 0. (3.3.22)
' V J\Xi\<Ck >
Let Bn = B^\ Then (3.3.18), (3.3.21) and (3.3.22) imply
Sn/Bn —у N(0, 1) as η —► oo.

3.3 The CLT and WIP when the variance is infinite
67
Now we turn to the weak invariance principle. Put ka = [k2+a]. For
any positive integer N there exists an η such that ka(p + q) < N < (ka +
l)(p + q). Define £j and 77j as above, however their indices are extended
to j = 0,1, · · ·, ka — 1. Recalling the proof of (3.3.18), under the condition
kaa(q) —► 0, there are constants Вт > 0 (га = /с, к + 1, · · ·, ка) such that
.. т—1
-£) Σ & Д ^(°> χ) asn^oo, (3.3.23)
where Sm^ are the normed constants related to the independent random
variables with the same distribution as £o (for fixed ρ ). If we denote the
sequence of constants corresponding to {£j, j = 0,1, · · -,ra} in Theorem
26.4 of Gnedenko and Kolmogorov (1954) by {Cm}, then, similarly to
(3.3.16),
B$2 = (1 + o(l))mp [ X\ dP. (3.3.24)
J\Xl\<C'm
And similarly to (3.3.19),
ka-i
ВЦ*'1 Σ Vj -^ #(0, 1) as η -> oo. (3.3.25)
j=o
Define Вдг = в£\ put
[(fca-l)*] [(fca-l)t]
5[^t] = Σ £? + Σ ^i + ^,*'
j=0 j=0
where 77^ = X([(fca_1)t]+1)(p+g)+1 + · · · + Х[лгф with the number of terms
[M]-(pa-l)*] + l)(p + g)<^
Define
[(fca-l)t] [(fca-1)*]
Recalling (3.3.17), we know that the normed sums of £j(vj) and the normed
sums of £/(77/), where £/, j = 1, · · · ,ka — 1, (77/, j = 1, · · · ,&a — 1) are
independent, with the common normed constants, have the same convergence.
Hence by Theorem 26.2 in Gnedenko and Kolmogorov (1954), (3.3.23)
implies that for any ε>0, 0 < £ < 1,
J\x\>e
l\x\>e

68
Chapter 3 Weak Convergence for α-mixing Sequences
[mt}{[ x2dF%\x)-([ xdF^(x)Y}^t,
where Fm\x) is the distribution of ξο/Bm, i.e.
[rut] J dF^(Vty)-^0,
[mt]{ f y2dF^(Vty) - ( / ydF^(Vty))2} - 1.
Then by the same theorem cited just now we obtain
W'N(t) Λ W(t). (3.3.26)
Similarly, we can also show
W'N(t) - W'N(s) Л W(t) - W(s) for 0 < s < t < 1. (3.3.27)
An analogue of (3.3.25) for {ijj} is
[(fca-l)t]
Σ Vj±W(t). (3.3.28)
В
(я)
'fca 3=0
With the help of the rusult similar to (3.3.20), (3.3.27) implies
uniformly in t. Imitating (3.3.22) we also have
p
Vnj/Bn -► 0
uniformly in t. Consequently
w" Д o.
Thus we investigate W'N instead of Wn-
At first we consider convergence of finit-dimensional distributions of
W'N. The convergence of one-diniensional distribution has been given by
(3.3.26). For the two-dimensional case we need to prove
(WM, W'N{t) - W'N{s)) Λ (W(a), W(t) - W(s)), 0 < s < t < 1,

3.3 The CLT and WIP when the variance is infinite
69
in other words, to show that for any Borel sets A\ and A2
P{W'N(s) € AX, W'N(t) - W'N(s) G A2)
-»· P{W(s) G A1}P{W{t) - W{s) G A2}. (3.3.29)
According to the α-mixing property, we have
\P{W'N(s) G Au W'N{t) - W'N{s) G A2)
- P{W'N(s) G A^PiW^t) - W'N(s) G A2}\
< <*(q).
Combining it with (3.3.26) and (3.3.27) yields (3.3.32). Three or more
dimension case can be treated in the same way, and hence the finite-
dimensional distributions converge properly.
Finally, we prove tightness of W'N. By (3.2.5), it suffices to show that
for any ε, η > 0 there exist a <5, 0 < δ < 1 and an integer щ such that for
0 < t < 1
P{ sup \WN(s)-W'N(t)\>e}<6^ n>n0.
t<s<t+6
Equivalently,
j
I
i=0
j
P{ max \У ij\ > εΒΝ) < δη, π> η0. (3.3.30)
By the CLT, we have
[2ka6]
p{\ Σ &| ^ εΒ»/ή - p{\ww)\ > ε/*}
i=o
= P{\W(1)\ > e/(2y/2S)} < ^-y/2SE\W(l)\3 < ^
provided that δ is small enough. Hence, there exists an no such that
P{\ Ε &| ^ SBN/2} <: Y> П^ П0-
i=0
Therefore, if we prove that for large N and every j, 0 < j < 2ka6 — 1,
ρ{\Σ&\<εΒΝ/ή>\, (3.3.31)
i=0

70
Chapter 3 Weak Convergence for α-mixing Sequences
it follows from Lemma 2.2.1 that we have
j [2ka6]
P{o<£f2Xj££| * eBN) < 2P{\ £ fc| > eBN/2} < δη.
г=0 г=0
Now it remains to show (3.3.31). First we consider the case of j < k.
Using (3.3.24) and noting ρ < Α;α, we have
^{|Σ&| > εΒΝ/ή < 2jpE\X1\/eBN
i=0
< ZkpElX^/eikaP [ XldP]1'2 -+ 0.
Assume к < j < 2ka6. Recall (3.3.23). By Theorem III, we can write
Bf=jh^(j).
With the help of the property of a slowly varying function,
2
Bf~ jh№(j)
hm sup —j- = hm sup , > < 26.
ЛГ-.00 Щ лг_оо kah(P)(ka)
Therefore, there exists an щ such that B2N/Bf] > 1/36 for п>гц. Let
6 be small enough so that
P{|W(1)| > e/2V36} < 1/4.
Then
j j
ρ{|Σ&| ^εΒ»/2} ^ ρ{|ί>| ^ ^Β^ν^δ}
i=0 i=0
-P{|W(1)|> e/2V36} < 1/4,
which implies (3.3.31). The proof of Theorem 3.3.1 is completed.

Chapter 4 Weak Convergence for
p-mixing Sequences
Ibragimov (1975) first showed that the CLT and the WIP hold true
under some conditions for a strictly stationary p-mixing sequence of random
variables, i.e.,
Theorem 4.0.1. Let {Xn,n > 1} be a strictly stationary p-mixing
sequence of random variables with EX\ = 0, EX\ < oo and
(i) a* = ES%^oo,
(η) Σ£°=ιΡ(2")<οο.
Then the distridution of Sn/an converges to Φ(χ).
Theorem 4.0.2. Let {Xn,n > 1} be a strictly stationary p-mixing
sequence of random variables with EX\ = 0,
Е\Хг \2+δ < oo for some δ>0
and condition (i) is satisfied. Then with Wn(t) = S[nt]/an, 0 < t < 1,
Wn=>W.
Peligrad (1982) gave a functional form under 2-th moment condition,
however, the mixing rate is more restrictive, namely Σρ1^2(2η) < oo.
Peligrad (1986) suggested five open problems, one of which is to prove the
weak invariance principle under the same sufficient conditions for the CLT.
Shao (1988b) gave a positive answer, which will be introduced in Section
4.1.
When the moments of order s > 2 is finite, Bradley (1984) raised the
problem of finding the slowest permissible /9-mixing rates to assure the
CLT under more flexible moment assumptions. Suppose that {Xn,n > 1}

72
Chapter 4 Weak Convergence for p-mixing Sequences
is a strictly stationary p-mixing sequence of random variables with EX\ =
0, σ\ — ES\ —► oc and EX\g[X{) < oc, where g : [0, oc) —► [0, oc) satisfies:
both g(x) and χ /g(x) are non-decreasing functions for some δ > 0.
Bradley asked whether the CLT holds true when
[logn]
exp(d Σ Р(*» = °(9(nl/2))
i=l
for any d > 0. Peligrad (1987) proved a more meticulous result.
Theorem 4.0.3. Let {Xn,n > 1} and g(x) be as above. If
[logn]
exp((2 + ε*) Σ p(?))=0(g(n^))
i=l
for' some о < ε* < 1, then Sn/an —► Λ^(0,1) in distribution as η —> oo.
Peligrad (1987) conjectured that Wn => W under the same conditions
as in Theorem 4.0.3. Shao (1989b) gave a positive answer, which will be
discussed in Section 4.2. In Shao (1989a), he gave a more general result
for the non-stationary sequence, which will be discussed in Section 4.3.
Some invariance principles for stationary /9-mixing sequences with infinite
variance will also be introduced in Section 4.4.
In the study of invariance principles for dependent random variables,
the following basic result due to Billingsley (1968) is used frequently. Let
В = σ{(—ос,ж), —ос < χ < оо}
be the Borel σ-field of R.
Theorem 4.0.4. Let {Wn,n > 1} be a sequence of random elements
in D[0,1] satisfying the following conditions:
(i) {Wn(t),0 < t < 1} has asymptotically independent increments,
i.e., for any given Bi G β, г = 1, · · ·, r and 0 < s\ < t\ < · · · < sr < tr < 1
we have
lim {P(Wn(ti) - Wn(si) G В{, г = 1, · · ·, r)
η—>οο
г
- Π P(Wn(U) - Wn(si) € Bi)} = 0,
i-1

4.1 The WIP when the momnets of order 2 are finite
73
(ii) {W%(t),n > 1} is uniformly integrable for each t,
(Hi) EWn(t) -> 0, EW*(t) -> t as η -> oo,
(iv) for any ε, η > 0 £/iere exisi α 5 > 0 and a positive integer no s^c/i
ίΛαί
P{w(Wn,6) >ε} <η n>n0,
where w(x,6) = sup|s_t|<<5 \x(s) — x(t)\. Then Wn => W.
4.1 The WIP when the moments of order 2 are finite
Shao (1988b) proved the following theorem.
Theorem 4.1.1. Let {Xn,n > 1} be a p-mixing sequence of random
variables with EXn = 0, EX2 < oo and
(i) limn-oo ESl/n = σ2 > 0,
(ii) {X2,n > 1} is uniformly integrable,
(Ш) Σ~=ιΡ(2")<οο.
Then
where
wn(t) = s[nt]/*v4 о < t < i.
The proof of Theorem 4.1.1 will need the following lemmas.
Lemma 4.1.1.(Peligrad 1982) Let {Xn,n > 1} be an α-mixing
sequence of random variables with EXn = 0, EX\ < oo, satisfying
(i) of Theorem 4-1-1 and
(ii/ for each t £ [0,1], {W%(t),n > 1} is uniformly integrable,
(iv) for any ε > 0 there exist a real number λ > 2 and a positive
integer щ such that for every η > no and all к > 1
P{ max \Sk(i)\ > λσ^/η} < ε/λ2. (4.1.1)
1<г<п
Then Wn => W.
Lemma 4.1.1 is a corollary of Theorem 4.0.4.
Lemma 4.1.2.(Moricz 1982) Let {Xn,n > 1} be a sequence of random
variables. Suppose that the non-negative function f(k,m) satisfies
f% m) + f(k + m, /) < /(*, m + l) (4.1.2)

74
Chapter 4 Weak Convergence for p-mixing Sequences
for fc>0,ra>l,/>l and the function g(t,s) is non-decreasing for each
arguments. If
E\Sk(m)\r < f(k,rn)gr(f(k,rn),rn) r > 1, (4.1.3)
then
EnSJSt(m)r < |/(*.„){ Σ »№·[?])}· <«·">
The proof of Lemma 4.1.2 will not be presented here.
Proof of Theorem 4.1.1. We need only to check the conditions
(ii)' and (iv) in Lemma 4.1.1.
1) We prove that {5|(n)/n,n > l,fc > 0} is uniformly integrable. Let
TV > 0 be specified later on. Denote
ΧτΝ = XiI(\Xi\ < N)-
k+n
S?in) = Σ, X?,
i=k+l
fc+n
^fc (n) = Σ Xi '
i=fc+l
£αί/= / UdP.
JU>a
It is obvious that
- EXiI(\Xi\ < Ν),
, С^^Ч
EaSl(n)/n < 4Ea/4(S?(n))2/n + 4E(Sk(n))2/n. (4.1.5)
Prom Lemma 2.2.2 it follows that for every η
sup E(S"(η))2/η < KsupE(X%)2
k>l k>l
for some К > 0. Since {X\,n > 1} is uniformly integrable, for any given
ε > 0 we have TV such that supfc>1 E(Xk )2 < ε/8Κ. Thus for each η > 1
and к > 1
£(sf(n))2/n < ε/8. (4.1.6)
On the other hand, from Lemma 2.2.4, there exists a constant K\ =
Κι(Ν,δ,ρ) > 0 such that for every η
supE\S?(η)\2+δ/n1+6'2 < Кг.
fc>l

4.1 The WIP when the momnets of order 2 are finite
75
Then for large a we obtain
Ea/A{S»{n)f/n < * Ea/A\S»{n)\^/n^l* < |. (4.1.7)
Inserting (4.1.6), (4.1.7) into (4.1.5) we prove that {S£(ra)/ra,ra > 1, к > 1}
is uniformly integrable.
2) We show (4.1.1). Let
ρ = [exp(2Clog1/3 n)], r = [n/p],
where the constant С is specified in Lemma 2.2.4 corresponding to δ = 1.
Denote
X? = XiI(\Xi\ < n^/p) - EXiI{\Xi\ < n^2/p),
k+l k+l
sj?(0 = Σ xf, sl(i) = Σ xj,
j=k+l j=fc+l
fc+2ir
У£= Σ X?, * = l,2,...,pi:=|p/2],
j=fc+l+(2t-l)r
fc+(2i+l)r
Zi= J; Χ?, ί = 0,1,.··,ρ2:=[(ρ-1)/2],
j=fc+l+2ir
Ti(i)= Σ ^' r,(i) = 5Jzj,
T(i) = T0(i), T(i)=T0(i).
It is obvious that Sk(i) = S%(i) + Sk(i). Without loss of generality assume
that σ — 1.
We first prove
Pi max |S£(i)| > λη1/2} < ε/6λ2 (4.1.8)
for large n. Form Lemma 2.2.4 with δ = 1 it follows that
max |£S£(i)|3 < C{n*'2al + nexp(Clog1/3n)a0}
l<fc<n
< cn3/2,
where σ^ = supn E(X\n^)2, ao = supn E\X\nχ. Using Lemma 4.1.2 we
have
sup£m^|S£(i)|3 = 0(n3/2).

76
Chapter 4 Weak Convergence for p-mixing Sequences
That is to say, {maxi<;<n |S£(i)|2/n,n > 1, к > 1} is uniformly integrable.
Therefore for any ε > 0 there exist λ > 2 and large no such that (4.1.8) is
satisfied for η > щ.
Next, we show
P{ max |s£(i)| > ЪХп1'2} < 5ε/6λ2. (4.1.9)
1<г<п
Note that the left hand side of (4.1.9) does not exceed
P{ max \T{%)\ > 2\n1l2) + p{ max \T(i)\ > 2Xn^2}
+ ^p{maxK+jr(i)|>An1/2}
j=o -г-т
=:/i+/2+/3. (4.1.10)
Since {X2,n > 1} is uniformly integrable, without loss of generality
we can assume that supn>1 ||Xn||2 < 1. Thus we have
P{max|5^ir«)l>An^}
fc+(i+i)r
<p{ Σ (\χ7\-ε\χ7\)>^2/2}
i=k+l+jr
Then by condition (ii) again
Now we estimate I\. Denote
ft = σ(Κι,· ··,*·), ui = E(Yi\gi-1),
Ui(i) = E£/+1 «,·, Tt(i) = Tt(i) - Щг),
U(i) = U0(t), T*(i) = T(i) - U(i).
It is easy to see that
h < P{ max |T*(t)| > An1/2)· + PJ max \U(i)\ > An1/2}
=:/π+/ΐ2· (4.1.12)

4.1 The WIP when the momnets of order 2 are finite
77
Since {Т*(г),г = 1, · · · ,ρχ} is a martingale, for large η
hi < ε/6λ2. (4.1.13)
In order to estimate /12, we prove that there exists a constant Co, which
does not depend on /,г, к and η, such that
М//2(г) < C0irp2(r)(\og2i)2. (4.1.14)
From Lemma 2.2.2 there exists a constant C\ > 0, which does not depend
on /, г, к and η, such that
ETf{i) < Сггг. (4.1.15)
Using induction for г, we can show that (4.1.14) holds for C0 = Ci/(log |)2.
From the definition of />-mixing, we have
EUf{l) = Euf+1 = EYl+1ul+1 < р(г)||У/+1||2|К+1||2.
Combining it with (4.1.15) implies (4.1.14) for г = 1.
For г > 2, assume that (4.1.14) holds for j < i. Put i\ = [г/2],гг =
г — i\. We have
М/г2(г) = EUf{h) + EUf+h{i2) + 2£;E7/(i1)E7/+il(i2)
<^(»1) + Щ2+<1(г2) + 2/1Кг)||С7,(»1)||2||ТЖ1(»2)||2.
By the assumption of induction and (4.1.15), we obtain
EUf{i) < C0i1rp2(r)(log(2i1))2 + C0i2rp2(r)(log(2i2))2
+ 2р2(г)гг1/2г2/2СУ2С11/21оё(2г1)
< C0irp2(r)((log(2i2))2 + 2(log^) log(2i2))
< C0irp2(r)(log(2i))2.
This proves that (4.1.14) holds for every i. From (4.1.14) and Lemma 4.1.2
we obtain
Ε max U2(i) < cPlr(\og(2pi))4p2(r) < cn(logn)~1/2.
1<г<рх
Thus for large η there exists a λ > 2 such that
J12 < ε/6λ2. (4.1.16)

78
Chapter 4 Weak Convergence for p-mixing Sequences
Combining it with (4.1.13) yields
h < ε/3λ2. (4.1.17)
By the same way we have
h < ε/3λ2. (4.1.18)
From (4.1.11), (4.1.17) and (4.1.18) it follows that (4.1.9) holds true. This
proves Theorem 4.1.1.
From Theorem 4.1.1 and Theorem 2.1.5, we have the following corollary
immediately.
Corollary 4.1.1. Let {Xn,n > 1} be a strictly stationary p-mixing
sequence of random variables with EX\ — 0, EX\ < oo and Σ^=ι Ρ(2η) <
oo. //σ2 = ES„ —► oo, then
η—►oo
Moreover if σ > 0, then
where Wn(t) = S[nt]/ay/n, 0<t<l.
4.2 The WIP when moments of higher than two orders
Let {Xn,n > 1} be a strictly stationary /9-mixing sequence of random
variables with EX\ — 0,EXf < oo and σ2 = ES„ —► oo(n —► oo). Let
g : [0, oo) —> [0, oo) be a non-decreasing function and for some 0 < δ <
1,χδ/g(x) be also a non-decreasing function.
Theorem 4.2.1.(Peligrad 1987, Shao 1989b) Let {Xn,n > 1} be as
above and satisfy
(i) EX1V|X1|)<oo,
(ii) exp((2 + ε*) T,t\n] p(2k)) = 0(g(n^))
for some 0 < ε* < 1. Then
Wn=>W.
By some simple computations we have the following corollaries:

4.2 The WIP when moments of higher than two orders
79
Corollary 4.2.1. Let {Xn,n > 1} be as above. Suppose that for some
ε > 0 and a > 0
EXf(log+iXil)2^1'^ < oo
and
p(n) < a/ log η
for every η sufficiently large. Then
Wn=>W.
Corollary 4.2.2. Let {Xn->n > 1} be as above. Suppose that for some
0</3<1,ε>0 and a > 0
ΕΧΪexp(^^(21°g+ Ι^Ι)1"') < °°
and
p(n)<a/(\ognf
for every η sufficiently large. Then
Wn=>W.
Corollary 4.2.3. Let {Xn,n > 1} be as above. Soppose that for some
r > Ο,ε > 0 and a > 0
^^9 / 4alog+ |Xi| \
and
p(n) < a/(loglogn)r
for every η sufficiently large. Then
Wn=>W.
The proof of Theorem 4.2.1 will need the following lemma, which is an
immediate consequence of Theorem 3.1.2 and Theorem 8.4 of Billingsley
(1968).

80
Chapter 4 Weak Convergence for p-mixing Sequences
Lemma 4.2.1. Let {Xn,n > 1} be as above. In order that Wn weakly
converges to W it is necessary and sufficient that {52/σ2,η > 1} is
uniformly integrable and for any given ε > 0 there exists α λ > 1 such that
P{ max \Si\ > λση\ < ε/λ2. (4.2.1)
Lemma 4.2.2.(Peligrad 1987) Let {Xn, η > 1} be as above with EX\ <
oo. Then for any given ε > 0 there exists а С = (7(ε,/9(·)) such that for
every η > 1
ES£<C(n1+eEXi + ali):
Proof. Denote am = ||5m||4. It is obvious that
d2m < \\Sm + 5fc+m(m)||4 + 2каъ
Using the Schwarz inequality and by the definition of p-mixing we have
£|Sm + Sfc+m(m)|4
< 2ai + 6E\SmSk+m(m)\2 + 8a2m(E\SmSk+m(m)\2)^2
< 2(1 + 7pl'2(k))a4m + 8a2ma2m + 6a4m
<(21/\l + 7p^2(k))1/4am + 2am)4.
It follows that
a2rn < 21/4(1 + lpl/2{k))llAarn + 2am + 2каг.
Let 0 < ε < 1/3 and к be large enough such that 1 + 7p1/2{k) < 2ε. By
the recurrence method for every integer r > 1 we have
r
a(2r) < 2Γ(1+£)/4αχ +2^2(i-1)(1+e)/4(a(2r-i) + ЫХ).
г=1
Whence
a(2r)<c(2r(1+£)/4a!+^(2r)).
This implies the conclusion of the lemma.
Proof of Theorem 4.2.1.
If Σρ(2η) < oo, it follows from Theorem 2.1.4 that the conditions of
Theorem 4.1.1 are satisfied. Therefore the conclusion of Theorem 4.2.1
holds true. We shall treat here the case when Σρ{2η) = oo. At this time

4.2 The WIP when moments of higher than two orders
81
we have that g(x) —► oo as χ —► oo by condition (ii). Without loss of
generality, we can assume that
p(n)>(logn)-x(loglogn)-2 (4.2.2)
for every large n.
1) We first prove that{5^/a^, η > 1} is uniformly integrable. It is easy
to see that the condition (ii) implies
[log n]
gin1/2) > exp(2 £ р(2*)/(1 - ε*)) (4.2.3)
г=1
for large п. Put
[(1-е) log η]
Τ = inv<?(exp(2 £ ρ(Τ)/(1 - ε*))), (4.2.4)
i=l
where 0 < ε < ε* < 1. Denote
ΧΛ = XiI{\Xi\ < Τ) - EXiI(\Xi\ < Τ),
Xi2 = XiI(\Xi\ > Τ) - EXiI(\Xi\ > Τ),
η η
г=1 г=1
σηΐ = Var5nb σ2η2 = VarSn2.
By Lemma 2.2.2 and noting that the function g(x) is non-decreasing, we
have
σ"2 <Ci^S*i2i7(l*i|)/(|*i| > T)
[(1-е) log n]
xexp( £ р(2<)/(1-е)),
г=1
where C\ — C\[e). Prom the definition of Τ and Lemma 2.2.3 we get
<& < ^olEX\g{\Xx\)l{\Xx\ > T). (4.2.5)
Obviously 'mvg(x) —► oo (x —► oo). It follows from (4.2.4) and (4.2.5) that
ση2 = ο(ση), (4.2.6)
which implies
<7ni/&n —► 1 as η —► oo. (4.2.7)

82 Chapter 4 Weak Convergence for p-mixing Sequences
By Lemma 4.2.2 for the sequence {Χηχ,η > l}, there exists a constant
K\ — Κι(ρ(-),ε) such that for every η > 1
E\Snl\4 < Кг(п1+е'2Т2ЕХ* + σ*ηΧ). (4.2.8)
From Lemma 2.2.3 and noting that exp(d^i=i р(2г)) is a slowly varying
function (n —► oo), it follows that there exists а С = С{р{-),е) such that
for every η > 1
σ4η > СП2'*'2.
Thus by (4.2.7) and (4.2.8) we have
E{\Snl\/anY <Κ2{Τ2/ηι-ε + 1),
where K2 = Κ2(ρ(·),ε).
From (4.2.3) it follows that for large η
[(1-е) log η]
5([nX-1X/2)>exp(2 £ р(2*)/(1-е*)).
г=1
Combining it with (4.2.4) we obtain that T/nSl~e^2 is bounded by 1.
Therefore
sup£(|Sni|/an)4<oo. (4.2.9)
η
Then, from (4.2.6) and (4.2.9), we prove that {5^/σ^,η > 1} is uniformly
integrable.
2) Next we show that for any ε > 0 there exists a λ > 1 such that
P{ max \Si\ > 6λση) < 6ε/λ2. (4.2.10)
Denote
/4n [bgn]
Τι =ΜΤ Σ Р2/{2+6)(*))> J = "1/2/7ъ (4.2.11)
г=1
Хг1 = Х{1(\Хг\ < J) - SXi/(|Xi| < J),
Xi2 = Хг/(|^г| > J) ~ EXiI(\Xi\ > J),
k k
Sni(k) = Y^Xn, Sn2(k) = y~]Xj2,
г=1 г'=1
a2nl(k) = ES2nl(k), a2n2{k) = ES2n2(k).

4.2 The WIP when moments of higher than two orders 83
Obviously Sk = Sni(k) + Sn2(k) and
P< max \Si\ > 6λση >
I l<i<n J
< P\ max \Snl(i)\ > λση\ + p{ max |5n2(i)| > 5λση}.
^1<г<п У 4<г<п У
We first note that
л п tlognl
1о§Т1 = f Σ Ρ2/(2+δ)(2Ί
Γ21
<f,-"(2+i'(^) Σ **)
Hence we have for every η sufficiently large
logT^fp-^^i^) £ p(2<) (4.2.12)
1 г=1
and
[log η] , [bg(n/T2)]
£ p(2') < (l + fr) Σ A**)· (4·2·13)
i=l i=l
Prom this and by condition (ii) and the fact that g(x) is non-decreasing
we have
9{J) - ехр(т^т^ Σ ρ(2% (4·2·14)
and by Lemma 2.2.2, Lemma 2.2.3 and (4.2.14) for every к < η and η
sufficiently large
[bgn] +
a2n2(k) < CkEXll{\Xx\ > J)exp( £ (l + -)/»(*))
s(J)
i=l
* [bgn]
<CC'-V^X125(|X1|)exp(1|- £ р(2*)).
г = 1

84 Chapter 4 Weak Convergence for p-mixing Sequences
From this and noting that Σρ(21) = oo, we deduce that
max = o(l) as η —> oo.
l<fc<n (Tk
Whence it is easy to see that for к = 1,2, · · ·, η and η sufficiently large
a2nl(k) <2σΙ (4.2.15)
By Lemma 2.2.4 and (4.2.15)
[log n]
E\Snl(k)\2+6 < c[al+6 ^ кЩХ^ЩХ,] < J)exp(30 £ ρ(2*))).
t=l
From this and by Lemma 2.2.2, conditions (i), (ii), (4.2.11), (4.2.14) and
Lemma 4.1.2 we see
Ε max \Snl(k)\2+s
l<fc<n
< c(a2n+e + n(logn)2+sE\X1\2+sI(\X1\ < J)
[log n]
x exp(30 £ рМ**>(2*)))
< c(o™ +
[log η]
x Cexp(35 ζ р2^+6\21)))
г=1
<ca2n+s(l + EX2g(\Xi\)).
Thus there exists a constant λ > 1 such that for every η sufficiently large
. r< max \Sni(k)\>Xan\<e/\2. (4.2.16)
We now estimate Р{тах!<^<п |5П2(^)| > 5λση}. Let
,50 [bsnl
i=l
2+s , n2+^(logn)2+^X12g(|X1|)
P = exP(y Σ Ρ2/(2+δ)(21)),
i=l
Г=Н' Л=[|], P2=[P_1
LpJ' *-* L2J' ~ L 2

4.2 The WIP when moments of higher than two orders
85
Put
Yi = Ej=l+(2t-l)r -Xj'2, * = 1, 2, · · · ,Pi5
Zi = Eflt+L XJ2, г = 0,1, · · · ,p2;
T1(i) = EUYJ> T2(i) = EUoZr
Noting that {Xj2,1 < 3 ' < τι} is stationary we have
P{ max |5n2(fc)| > 5λση [
< p{ max |2i(fc)| > 2\σΛ
+ p{ max |T2(fc)| > 2\σΛ
+ (p + 1)P{ max |Sn2(fc)| > λση}
=: /χ + /2 + /3·
In terms of Lemmas 2.2.2 and 2.2.3 we have for every η sufficiently large
p{ max \Sn2(k)\ > λση\
{l<k<r *
< p{J2(\Xn2(i)\ - E\Xn2(i)\) > λση - 2^ E\Xn2(i)\}
г=1 i=\
< ρ{έ(ΐ^(οι - тъ*т > λση - 2r^ffll}}
< р{£(|Хп2(г)| - E\Xn2{i)\) > \ση/2]
г=1
[log r
<4Cra-2exp((l + e74) £ ^^/(l^il > J) · λ"2.
г=1
Whence by (4.2.14)
[log η]
/3<4CW-2exp((l + £*/4) £ р^ЯХ2/^! > J) · λ~2
г=1
4Г / [logn]
- c^(7)exp((2 + 3e*/4) Σ Р(^))^^(1^1|)
AC / p* [bgnl \
г=1
< ε/λ2. (4.2.17)

86
Chapter 4 Weak Convergence for p-mixing Sequences
In order to establish the estimation of /χ, let
TQ = (Ω, 0), Tk = σ(Χ» 1 < i < 2rfc);
uk = E{Yk\Tk-l), Λ=1,2,···,ρι;
i+k
Щк)=^иа, Т*(к) = Т1(к)-и0(к).
Obviously
lx < P{ max \T*(i)\ > λση\ + PJ max |E70(i)l ^ λση|
=■ hi + /ΐ2·
Because {Τ*(г), г = 1,2, · · · ,ρχ} is a martingale, we have
n г=1
In a way somewhat similar to the estimation of /3 we also have for any
λ > 1 and for every η sufficiently large
hi < e/λ2. (4.2.18)
Finally, we shall prove by induction on к that for every г, /с, η,
EU?(k) <Ckp2(r)log2(2k)EXfl(\X1\ > J)
[log n]
•r-exp((l + e74) £ p(2')). (4.2.19)
г=1
When A; = 1, by the definition of p-mixing
EU?(1) = EE2{Yi+x\Fi) = E(Yi+1E(Yi+1\Ti))
<p(r)\\Yi+1\\2-\\E(Yi+1\^)\\2,
thus (4.2.19) is true for к = 1 and for every г + 1 < p\ by Lemma 2.2.2.
When к > 2, assume (4.2.19) holds for every integer less than k. Put
kx = [fc/2], A?2 = & — fci, then
tft/2(fc) = M/?(fcx) + EU?+kl(k2) + 2EUi{ki)Ui+kl(k2)
i-\-k
= EUhh)+EUi+kl(k2) + 2EUi(k1) Σ Yj
j=i+ki + l
<EU?(ki) + EUf+kl(k2)
i-\-k
+ 2||tfi(*1)||2.|l Σ Yihp{r).
j=i+ki+l

4.2 The WIP when moments of higher than two orders 87
By induction hypothesis and Lemma 2.2.2
EUf{k) < C{kx log2 2fci + k2 log2 2k2 + 2(^2)1/2 log 2kx)
* [losn]
• p\r) ■ r ■ exp((l + γ) Σ Р(2»")).Е7АГ?7(|ЛГЖ| > J)
// e*x [1°s"] \
< Cfc(log2fc)2r · exp((l + T) Σ ^))
•EXf/dXx^ J).p2(r).
which proves that (4.2.19) holds.
Prom (4.2.19) we obtain by Lemma 4.1.2
Ε max t/2(i)
1<г<р1
'2(
<3Crp1/9J(r)log4(2p1)
[log n]
•exp((l + e74) Σ Pi^B^/dXx^ J)
i=i
3Ca^2(nM)log4(2Pl)
C"i/(J)
[log n]
.exp((2 + 3s*/4) £ Р(2*))Я^(|Х1|)
3Ca2p2(nM)log4(2Pl)
<
<
С
* [1оёп1
βχρ(-^ Σ /^^νιχιΐ).
By (4.2.12)
Fn [log n]
<Μ*.>*(τ)>(ΐ?)Σ*').
hence we finally get that for any λ > 1 and for every η sufficiently large
/12 < ε/λ2.
Therefore
h < 2ε/λ2. (4.2.20)

88
Chapter 4 Weak Convergence for p-mixing Sequences
Similarly, we have
h < 2ε/λ2. (4.2.21)
(4.2.10) now follows from (4.2.16), (4.2.17) and (4.2.20), (4.2.21). Theorem
4.2.1 is proved.
4.3 A generalized result when moments of higher
than two orders
Shao (1989a) gave a generalized result of Theorem 4.2.1, where the
condition of strict stationarity was removed.
Theorm 4.3.1. Let {Xn,n > 1} be a p-mixing sequence of random
variables with EXn = 0, EX\ > с > 0. Suppose that g : [0, oo) —► [0, oo) is
a non-decreasing function and χδ /g(x),0 < δ < 1, is also non-decreasing.
If the following conditions are satisfied:
(i) {X^g(\Xn\)^n > 1} is uniformly integrable,
(ii) σ\ \— ES\ = n/i(n), where h(n) is a slowly varying function,
(Hi) sup^o^! ΕΞ^τή/σΙ < oo,
(iv) lim^oo infm>0 ES^{n) = oo,
(υ) Βχρ[(2 + εηΣΪΛη]Ρ(^)) = 0(<?(n1/2)),for some 0 < ε* < 1,
then Wn weakly converges to W.
The proof of Theorem 4.3.1 needs the following lemmas.
Lemma 4.3.1 . Let /(A:, m) be a non-negative function satisfying (4-1.2).
Suppose that there exist a > 0, r > 1 such that
E\Sk(l)\r < f(k,l){fa(k,l)Wl(f(k,l)) +w2(f(k,l))}, (4.3.1)
where W2 is a non-negative non-decreasing function, w\ is a non-negative
function with
max si3w1(s) < alPwx{t) (4.3.2)
0<s<t
for some a>0,0 < β < a and any t > 0. Then we have
Ε max \Sk(i)\r
1<г<п
< 2r+1f(k,n){a(l - {\)(a-^rrfa{Kn)wx{f{k,n))
+ w2(f(k,n))logr(2n)}. (4.3.3)

4.3 A generalized result when moments of higher than two orders 89
Proof. From (4.1.2) and (4.3.2), we have for / < η
f(k, /W/(fc, /)) < af(*(k, n)Wl(f(k, n)).
Therefore, by (4.3.1), for every к > 0,1 < / < η
E\Sk(l)\r < f(k, 1){/α~β(ΚI) ■ af0(k,n)Wl(f(k,n)) + w2(f(k,n))}.
It follows from Lemma 4.1.2 and the monotonicity of W2{·) that
Ε max \Sk{l)\T
l<l<n
[logn]
1 i=0
<^f(k,n){w12/r(f(k,n))log(2n)
+ (ar(fc,n).1(/(fc,n)))^(l-(i)(a-/3)/71}r
<5.2r-2f(k,n){ar(k,n)Wl(f(k,n))(l - (λγα-β)/τγΓ
+ w2(f(k,n))\ogT(2n)}.
This completes the proof of Lemma 4.3.1.
Lemma 4.3.2. Let {Xn, η > 1} be a p-тгхгпд sequence with EXn = 0,
and <?i,<72 > 2. Suppose that the non-negative function h(n) satisfies:
max(fc(g]),fc(n- [£])) < 0h(n) (4.3.4)
χ 2_
for every η > 1 and some 0 < θ < 2 *ιΛ3, and if q\ > 3
[logn]
Λ(»)>-βφ{-αΣ ρ2/9ι(2Ί}, (4.3.5)
г=0
max iah(i) < anah(n)
Ki<n

90 Chapter 4 Weak Convergence for p-mixing Sequences
for some a > 0 and some a, 0 < a < q\ — 2. And suppose that integers
1 < к < η, I > 0 and numbers χ > 0,0 < В < A < oo satisfy
4n max E\Xi\I(\Xi\ > A) < x, (4.3.6)
l<i<l+n
Ш max E\Xi\I(\Xi\ > B) < x, (4.3.7)
l<i<l+n
j+n
Varf V XiI(\Xi\ < B)) < nh(n) max ΕΧ?Ι(\Χ{\ < В).(4.3.8)
2=7 + 1
TTien for any given ε > 0, 2/iere e:mte а К — Κ (ε, #ι, #2, α? 0, α, ρ(·)) such
that
р{тах|5Кг)|>х}
< 2 ρ(ΐ^ι > ^)
+ tfjaT91 Г(пй(п) max EXp{\X{\ < B))qi^2
{ L l<i<l+n
[log n]
+ nexp{tf ζ PVqi(2i)}^gqi(2n)
г=0
x max ЯШ^/ПХ;! < B)}
i<i<i+n ' ' Vl ' — yJ
[log n]
+ x-"2[(nexp{(l + e) £ р(2{)}
x max EXfl(B <Щ< A))42'2
l<i<l+n '
[log n]
+ nexp{# ]Γ p2/<?2(2*)}
χ max £|Х^2/(5< |X;| < А)]
l<i<l+n J
[log n]
+ Ж-2пр2(А;)1о§4ф-ехр{(1 + в) £ p(2<)}
г=0
x max EX?I(В <\Х{\<А)\.
l<i<l+n >
Proof. For simplicity, we assume that Χη-,η > 1 have a common

4.3 A generalized result when moments of higher than two orders
91
distribution. Denote
Xh = XiI(\Xi\ <B)~ EXiI(\Xi\ < S),
Xi2 = XiI(B < \Xi\ <A)- EXiI(B < \Xi\ < A),
Xi3 = XiI(\Xi\ > A)-EXiI(\Xi\ > A),
l+i
Slm(i) = Σ X3^ Ш = 1'2'3·
3=1+1
It is easy to see that
P{ max |5,(t)l > *}
1<г<п
< P{ max \Sn(i)\ >j} + P{ max |5i2(t)| > j}
1<г<п 4 1<г<п 4
+ P{m^|5i3(i)|>f}
1<г<п Ζ
=: h + h + h- (4.3.9)
By (4.3.6), we have
h < p{ Σ тпш > *) > \ - Σ вдм-ы > *)}
г=/+1 г=/+1
< P{ Σ \Xi\I(\Xi\ >A)>j}
i=l+l
l+n
< Σ Р(Ш Ζ Α). (4.3.10)
г=/+1
By Lemma 2.2.6, there exists a K\ = Ki{qi,a,9,p(-)) such that
*| Σ Ц
i=j+l
< Kx^nhMEXfldXxl < B))^'2
[logn]
+ nexp[#i Σ p2fqi(2i)\E\X1\qiH\X1\<B)}.
г=0
It follows from Lemma 4.3.1 that there exists a K2 = K2(Ki,a,qi) such

92 Chapter 4 Weak Convergence for p-mixing Sequences
that
Emax\Sn(t)\*
1<г<п
< K2{{nh{n)EXll{\X1\ < B))41I2
[bgn]
+ nexp{K2 Σ Р2/Я1(?)}
г=0
■(log(2n))^E\X1\^I(\X1\<B)}.
Then we have
h < K2x~qi{(nh(n)EXfl(\X1\ < B))qil2
[log n]
+ пехр{к2 Σ p2/«(2i)}(log(2n))<'1
i=0
■ElXi^IdX^KB)}. (4.3.11)
In order to estimate /2, put
(2i+l)fc+Z
j=2ifc+Z+l
2(i+l)fc+Z
^г = ]T Xj2, ί = 0,1,···,ρ2,
j=(2i+l)k+l+l
t
where pi = [(^ - 1)/2]>P2 = [(£ - 2)/2]. Denote
j=Q j=0
It is easy to see that
h < P{ max \Wi\ > 41 + P{ max |W*| > —)
1 i+i 1 * ι
+ Ρ·{ max max V^ Xv2 > — f
lo<i<[n/fc] ifc+i<i<(i+i)fc I l/=/^fc+1 I 12 J
=:/21 +/22 +/23· (4.3.12)

4.3 A generalized result when moments of higher than two orders 93
From the condition (4.3.7) and Lemma 2.2.5 it follows that
Z+(t+l)fc
'"^•Жч^ Σ}\Χ,\'(Β<\Χ,ί<Α)
-Ε\Χΐ\Ι(Β<\Χ3\<Α))>^
i+(t+l)fc
-2 Σ Ε\Χι\Ι(Β<\Χι\<Α)}
j=l+ik+l
i+(i+l)fc
Λ 0<г<[п/к] (j=l+ik+1
- E\Xj\I(B < \Xj\ < A)) > ж/24}
[bgfc]
< С^х-ъ{(кехр((1 + е) ]Г р(2*))ЕХр(В < \Xi\ < A))
i=0
[bgfc]
+ kexp{c Σ pVv^ElX^IiB < \X\ < A)}
i=0
[logn]
< Cx-q2{(nexp{(l + e) £ р(2')}яХх2/(Я < |Χχ| < Л))
i=0
[logn]
+ nexp{c Σ ρννφήΕΐΧ^ΙίΒ < \Χχ\ < A)}.
i=0
Next, we estimate /21. The estimation of /22 can be obtained by the same
way. Denote T-\ = {0, Ω}.
Ti = a(Xj :j<l + (2i + l)k), г = 0,1, · · · ,ρχ.
Put
Ui = Y% — i?(Yi|.Fi_i), Gj = Σ)=ο Uj,
Н{ = Σ)=ι Я(ВД--1)> i = 0,1, · · · ,Pl.
It is easy to see that
hi < P{ max \Gi\ > ж/24} + P{ max |#;| > ж/24}
=: /21(1) + /2i(2). (4.3.13)
Since {Ui^iyi > 1} is a martingale difference sequence, by the maximum

94
Chapter 4 Weak Convergence for p-mixing Sequences
value inequality of Brown (1971), we have
24 χ
hi(l)<-E\GPl\I(\GPl\>-)
<f{E\WPl\l(\WPl\>^)
+ E\HPl\l(\HPl\>^)}
< (эб/хря^!92 + (96/χ)2 е\нР1\2.
Prom Lemma 2.2.5 it follows that
E\WPir
[log n]
<c{(nexp[(l + s) Σ Р{У)]ЕХ11{В <\XX\ < A))
(4.3.14)
92/2
i=0
[log n]
+
nexp[c £ р2/*(2*)]^|Х1|да/(В < |Xi| < Л)}. (4.3.15)
i=0
We prove below by induction that there exists a constant K' such that
г+m 2
s( Σ ВД|^·-!))
< K'mkp2{k){\og{2m))2
[log m]
x exp[(l + e) ]T p(2j)]^Xi/(S < |Xi| < Л). (4.3.16)
i=o
Indeed, it follows from Lemma 2.2.1 that there exists a constant С such
that for mk < η
г+т
[log η]
E{ Σ yi) <C"mfcexp[(l + e) £ p(2^)
i=j+i i=o
χ EXfl{B <\Xi\< A). (4.3.17)
When m = 1, by the definition of p-mixing
E(E(Yi+1\Fi))2 = E(Yi+1E(Yi+1\Ti)) < p(fc)||yi+1||2||£;(yi+1|^i)||2,
that is
E(E(Yi+1\^))2 < p2(k)EY2+v
Let K\ = C'l log2(3/2), it follows that (4.3.16) holds true for m = 1.

4.3 A generalized result when moments of higher than two orders
95
Suppose that (4.3.16) holds true for the integers less than m. Now let
us show that (4.3.16) holds true for m. Denote mi = [πι/2],πΐ2 = m — m\,
we have
i+m
i+m\
= ε(Σ Ε(χ№-ι)Υ + ε{ Σ Е{у,\ъ-х)У
j=t+l j=i+mi + l
i-\-m\ i-\-m
+ 2Ε(Σ E(Yj\Tj_1))( £ Я(ВД-1))
j=i+l j=i+mi+l
г+mi 2 г+т
<Ε(Σ ВД^-ι)) +E( £ Я(ВД_х)У
i+m\ 0 г+т
j=i+l j=i+mi + l
г+mi г+т
+ 2ρ(Ι<)Ε\\Σ Е(Ъ\Ъ-1)Ц Σ yi
which, by the inductive assumption and (4.3.17), implies
г+т
e( £ е<у№-х)У
< К'{mi log2(2mx) + m2 log2(2m2)
О
+ 2 (log -)(m1m2)1/2log(2m1)}
[logn]
x p2(k)kexp{(l + e) ^ p(2j)}£X2/(# < |Xi| < A)
i=o
< К'ткр2(к) log2(2m)
[logn]
x exp{(l + e) £ р(27')}^Хг2/(5 < |Χι| < A).
j=Q
This proves (4.3.16). Moreover, it follows from (4.3.16) and Lemma 4.3.1
that there exists a constant K" such that
[logn]
ЕтяХ1Н2<К"р1кр2(к)(1оЕ(2р1))4ехр{(1 + £) £ p(2>')}
-г-Р1 j=o
• EXfl(B < \Χχ\ < A). (4.3.18)

96
Chapter 4 Weak Convergence for p-mixing Sequences
Thus we have
/21 (2) < 288K"x-2np2(k)log4[n/k]
[log n]
•exp{(l + s) Σ p(2j)}EX*I(B < \Хг\ < А). (4.3.19)
From (4.3.13), (4.3.14), (4.3.15), (4.3.18) and (4.3.19) it follows that
[log n] .
/21 < K3x-v{(nexp{(l +e) £ P(V)}eX2I(B < \Χχ\ < A))"2
i=o
[logn]
+ nexp{^3 Σ р2/*(2>')}ВД|да/(В < \Хг\ < A)}
3=0
+ К3х-2пр2(к)1оЕ4[^\
[log η]
x ехр{(1 +ε) £ р(У)}еХ*1(В < \Хг\ < А)
for some K% > 0. The proof of Lemma 4.3.2 is completed.
Lemma 4.3.3. Let 0 < δ < 1. Suppose that the non-negative function
h(n) satisfies the following conditions: there exist integer no > 0, 0 < θ <
2δ/(2+δ)^ о <s' <δ and a > 0 such that for any n>n0
h(ffl)wh(n-ffl)-ehin)' <4·3·20)
max i6 h{t) < an6 h(n). (4.3.21)
1<г<п
Let {Xn,n > 1} be a p-mixing sequence with EXn = 0,EX% < oo and for
every k > 0, η > 1
ESl(n) < nh(n) max EXf. (4.3.22)
k<i<k-\-n
Then for any given ε > 0 Йеге exists α Κ = Κ(ε, <5,5 , no, α, ρ(·)) змсЛ. that

4.3 A generalized result when moments of higher than two orders 97
for any η > к > 0, Ζ > 0 and В > О
Ε тах|5/(г)Г/( max \Si(i)\ > x)
1<г<п 1<г<п
[log η]
<K{x-8Unh{n) max EX? + nexp{(l + ε) V ,9(2*)}
^ ^ l<i<l+n { τ~~ί '
i=0
x max EXfl(\Xi\ > B))^
l<i<l+n
[log n]
+ nexp{K Σ p2/(-2+s\2i)}\og2+s(2n)
i=0
x max E\Xi\2+sI(\Xi\ < B)\
l<i<l+n >
[log n]
+ nexp{(l + e) £ p(2*)}(l + p2(fc) log4[J])
max £X?/(|XJ > 5)
i<i<i+n ~~
+ nfc max (£|Χ;|/(|Χ;| > Б))2). (4.3.23)
Proof. Denote
l+i
Xix = XiI(\Xi\ <B)- EXiH\Xi\ < B), Sn(i) = Σ хзъ
3=1+1
l+i
Xi2 = XiI(\Xi\ >B)~ EXiI(\Xi\ > S), 5/2(г) = £ Xj2.
3=1+1
It is easy to see that
Ε max Sf(i)I[ max |5/(г)| > ж)
1<г<п \1<г<п /
< 4£ max 5/21(г)/( max |5л(г)| > ж/2) +4Е max S^i)
1<г<п 1<г<п 1<г<п
< 8x~sE max |5/г(г)|2+г +4Е max Sj^t)
1<г<п 1<г<п
=:8/х+4/2. (4.3.24)
From Lemma 2.2.2 , for every η > 1, Ζ > 0 we have
[log η]
Я5?2(п) < Cnexp{(l + ε) £ р(2{)} max £X2/(|Xi| > В),
^ .—* J 1<г<1+п
г=0

98
Chapter 4 Weak Convergence for p-mixing Sequences
where С = С (ε). Therefore, by (4.3.22) one obtains
ESf^n) < 2ESf(n) + 2ESf2(n)
< 2nh(n) max EX?
~ ' l<i<l+n
[log n]
+ 2Cnexp{(l + s) Σ ρ(Τ)}
t=0
χ max EXfl(\Xi\ > B). (4.3.25)
l<i<l+n
It follows from Lemma 2.2.6 that there exists a K\ such that for every
l>0,n> 1
E\Sn(n)\2+s
<ΚΛ (nh(n) max EX2
~ IV v 'i<i<i+n l
[10ёП] - Ч(2+«)/2
+ nexp{(l + £) £ p(2')} max ЕХ?1(\Х{\ > B)Y
*■ T~T ' l<i<l+n /
г=0 ~~
[log n]
+ nexp{K1 У] ρ2/(2+δ)(2*)| max ВД|2+^(|*;| < В)}.
*■ Г~Т J /<г</+п У
г=0 ~~
By Lemma 4.3.1 we get
h < x~6c\ (nh(n) max EXf
[log η] 2±6
+ nexp{(l + e) £ р(2*)} max ЯХ2/(|Х;| > В)) 2
г=0 ~~
[log n]
+ nexp{(l + £) £ р2«2+6\Г)}
г=0
• max E\Xi\2+sI(\Xi\ < B)\og2+s(2n)\. (4.3.26)
l<i<l+n >
We estimate /2 below. Denote
Χ = ς£/+«&*;2, wi = EJ=on·, » = ο,ι,·.·,Ρι,

4.3 A generalized result when moments of higher than two orders
99
where pt = [(£ - l)/2], p2 = [§] - 1. We have
I2 < 8ΪΕ max Wf
~ l 0<i<pi %
l+j Ι2Ί
+ Ε max W*2 + E max max / Xv2 г
o<i<P2 o<t<[n/fc]tfc+i<j<(t+i)fcl f-f, , I J
=:8(/21+/22 + /23). (4.3.27)
It is easy to see from the proof of Lemma 4.3.2 that
[log n]
hi + /22 < с nexp{(l + ε) ]Γ p(2f)}
г=0
• (l +p2(A;)log4[Jl) max ЕХ21(\Х>\ > β). (4.3.28)
^ LACJ / 1<г<1+п
And from Lemma 2.2.2
i+i
hz < У Ε max У" Χυ2
о<г<[п//е] τ -./-ν -τ / -y=/+2fc+i
Z+(t+l)fc
<32 Σ И Σ {ΐ^|/(|χ«ι>5)
0<i<[n/fc] v=Z+ifc+l
-Я|Х„[/(|Х„|>В)}|2
+ k\ max (ВД/(|Х,|>Я)2)
[log n]
<c{nexp{(l + e) £ p(2f)} max £Хг2/(|Х;| > S)
г=0 ~~
+ nk max (£|X;|/(|Xi| > В))2}. (4.3.29)
l<i<l+n J
Inserting (4.3.26)-(4.3.29) into (4.3.24) we prove (4.3.23), as desired.
Proof of Theorem 4.3.1.
By the condition (ii), we need only to show that
(a) {S?tJa^n > 1} is uniformly integrable for any 0 < t < 1.
(b) For any given ε > 0, there exist a λ > 1 and an integer no such
that for η > no, 0 < к < ηλ2/ε we have
fc+г
Σ
fc+г
P{ max I V iJ > λση| < ε/λ2. (4.3.30)

100
Chapter 4 Weak Convergence for p-mixing Sequences
To this end, we need only to show that for any given ε > 0 there exist
a λ > 1 and an integer no such that for every η > no, / > 0
Ε max Sf(i)I( max |S/(i)| > X(nh(n))^2)/nh(n) < ε. (4.3.31)
1<г<п 1<г<п
Without loss of generality we assume that for η > 16
p(n) > l/(logn(loglogn)2). (4.3.32)
In fact, if put p*(n) = p(n) V (logn)_1(loglogn)~2, it is easy to check that
p*(n) satisfies condition (v) also.
By conditions (i), (ii), (iii) and Lemma 4.3.3, there exists a constant
К such that for every n, l<fc<n, В > 0 and λ > 0, we have
Ε max S,2(i)/( max |5/(г)| > X(nh(n))^2)/nh(n)
1<г<п 1<г<п
[log n]
<^{({n/i(n) + nexp((l+ £-) £ p(2<))
г=0
о 2+6
χ max EXfl(\Xi\ > B)}~
1<г<1+п
[log n]
+ nexp(tf Σ p2l{-2+8\2i)){\og{2n)f+e
2+6 .
Put
χ max E\Xi\2+eI{\Xi\<B))/{\s(nh(n))2)
l<i<l+n / /
[log n]
+ nexp((l + |) £ /»(*)) /<nu«n^?/(|Xi| > 5)
x(l+p2(A:)log4[^])M(n)
+ nfc max (£7|Xi|/(|X<| > B))2/nh{n)\
l<i<l+n )
=: X(/i + /2 + /3). (4.3.33)
г=0
Б = η1/2/?1, к = [η/Τ2] + 1.

4.3 A generalized result when moments of higher than two orders 101
We first estimate I\. Note that
8 к
г=0
+ ^Ρ2/{2+δ)φ Σ ι
t=[lognT-2]+l
[lognT-2]
<3*р-*/<**)(£) Σ ρ(2*)
г=0
+ 7Лру(ы)ф1оёт.
Therefore for large η we have
о 7^ [bg ηΤ-2]
10ёТ^х(1 + Й)^Д2+г)(^) Σ **) (4-3.34)
г=0
and
[log n] [log пГ~2]
Σρ(2*)<(ΐ + ^) Σ ρ(ή (4.3.35)
г=0 г=0
By condition (v), there exists a Ci > 0 such that for every η > 1
[log n]
<7(n1/2) > Cx exp{(2 + ε) Σ ρ(2<)}. (4.3.36)
t=0
Combining it with (4.3.35) and condition (v), we obtain
5(5) > d expfj-^ Σ ***)} (4-3.37)
' г=0
for large n. By Lemma 2.2.3 and condition (iii), there exists a constant
C2 > 0 such that for large η
[log η]
ESl>C2nexp{-(l+£-) £ p(2')},
г=0

102 Chapter 4 Weak Convergence for p-mixing Sequences
and hence, by condition (ii), we have
[log n]
h(n) > C2exp{-(l + -) Σ р(2>)}. (4.3.38)
г=0
By the monotonicity of g(x) and х^/д(х), we obtain
maxEXfl(\Xi\ > B) < -^-max ΕХ2д(\Х{\)I(\Xi\ > ^(4.3.39)
г>1 9\£>) *>1
B6
тахВД|2+*/(|Х;| < В) < ——тъхЕХ?д(\Хг\). (4.3.40)
г>1 9\Н) *—*
Combining (4.3.37), (4.3.39),(4.3.40) and (4.3.32) together implies
[log η]
nexp{(l + i) Σ р(21)}тяхЕХ?1(\Х>\>В)
г=0 ~~
< ^maxEXfgilXmiXil > B), (4.3.41)
C1C2 *>1
[logn]
nexpiK Τ p2>V+6\2i))\og2+\2n)mbxE\Xi\2+8I{\Xi\ < В)
[log η]
<nexp{2K Σ Р2/{2+6)(^)}в6т^ЕХ!д(\Х{\)/д(В)
г=0
[logn]
<п(2+6У2тахЕХ?д(\Х>\)ехр{-К Т. р2/(2+6)(2*)}
< (п/г(п))(2+г)/2тах^Х^(|Хг|) (4.3.42)
г>1
for large n. We have тахг>1 ЕХ?д(\Х{\) < оо by condition (i). Therefore
we obtain
h < е/(ЪК) (4.3.43)
for large η provided that λ is large enough.
We now estimate /2. Prom (4.3.34) and the definition of /с, we have
(1оёф)У(А0<2(1о§Г)У(А:)
*(£)Vli)V£>r*(1:"W
<3K\*
i=0
[log n]
£ (χ) ( Σ **>) · (4.3.44)
6 i-o

4.3 A generalized result when moments of higher than two orders 103
By (4.3.44),(4.3.39),(4.3.37) and (4.3.38), we get
[bg я]
4' ,=0
maxEXfl(\Xi\ >B)(l + p2(k)log4[£])
ι / \ [l°gn]
g(B)h(n) "U i, ыо
• (l + (χ)4( Σ Ρ(2')) ) maxEX^d^D/d^l > B)
ι/ \ [bgn]
^Μ-ίΣ^)}
г=0
от>- 4 [bgn]
• (l + (χ) ( Σ Ρ(*)) ) тах^Х^(|Х,|)/(|Х{| > В)
г=0
< спЛ(п) maxBX^dXiD/dXil > В), (4.3.45)
г>1
where the following result is used:
[bgn] [logn]
(Σρ(2*)) βχρ{-ϊ Σ^)} = °(1)·
г=0 г=0
Combining (4.3.45) with condition (i), we have
h < e/ZK. (4.3.46)
Finally, we consider /3. Note that
v^E\Xi\H\Xi\ >B)< щщтжЕХ?д{\Х№\Хг\ > В).
Hence it follows from (4.3.37) and (4.3.38) that
knmax{E\Xi\I{\Xi\ > B))2
г>1
< 2-^™*{Exh{\xm\Xi\ > в)?.

104
Chapter 4 Weak Convergence for p-mixing Sequences
By condition (i), we have
h < e/3K (4.3.47)
for large n. Inserting (4.3.43),(4.3.46) and (4.3.47) into (4.3.33) we prove
(4.3.31). The proof of Theorem 4.3.1 is completed.
From Theorem 4.3.1, we have the following corollary immediately.
Corollary 4.3.1. Let {Xn,n > 1} be a strictly stationary p-mixing
sequence of random variables with EX\ = 0, EX\ < oo, σ\ — ES\ —> oo.
If one of the following conditions is statisfied:
(i) EXlg(\Xi\) < oo, and for some 0 < ε < 1,
[log n]
exp((2 + s) £ ρ(2ή) = OMn1'2)),
k=l
(ii) for some ε > Ο,α > 0, EXl{\og |-^ι |)2α/(1_ε) < oo and χ ρ{η) <
a/logn,
(Hi) for some 0 < β < 1, ε > 0, a > 0,
£;X2exp{Ml+£)(2iog|Xl|)i-/S} < oo
and
p{n) < a/{\ognY,
(iv) for some r > 0, ε > 0, a > 0,
r2 /4a(l + e)log|Xi|
EX{ exp —ул—Ί '* ' ' ) < oo
1 ^V (logloglXxl)- J
p{n) < a(loglogn)"r,
then
4.4 The WIP when the variance is infinite
Bradley (1988) established a CLT for a strictly stationary /o-mixing
sequence with infinite variance. Shao (1993a) showed a WIP under the
same hypothesis.

4.4 The WIP when the variance is infinite
105
Theorem 4.4.1. Let {Xn,n > 1} be a strictly stationary p-mixing
sequence of non-degenerate random variables with EX\ = 0. Suppose that
(i) H(x) := ΕΧιΙ(\Χι\ < x) is slowly varying as χ —► oo,
(it) p(l)<l,
(iii) Σ~=ιΡ(2ί1)<οο.
Then there exists a sequence {An, η > 1} of positive numbers with An —> oo
as η —> oo, such that
Wn=>W
where Wn(t) = S[nt]/An, 0 < ί < 1.
Remark 4.4.1. In fact, Shao (1993a) showed a more general result
as follows: Let g : (—oo, oo) —► [0, oo) be a non-decreasing continuous even
function and χδ/g(x) is non-decreasing for any δ > 0 and ж large enough.
Let
[log x] [log x]
e(x,e) = exp{e £ р(2*)}, a(6) = exp{ £ p1"5^)}.
г=1 г=0
Suppose that conditions (i) and (ii) in Theorem 4.4.1 are satisfied and
(i)x G(x) := EXfg(Xi)I(\Xi\ < x) is slowly varying as χ —► oo,
(iv) G{x)e{x2,2 + ε) = 0(H(x)g(x)) for some 0 < ε < 1,
(ν) g(x) = 0(g(x/x(6))) or G{x) = 0{G{x/x{6))) for some 0 < δ < 1
as ж —► oo.
Then we also have the conclusion of Theorem 4.4.1.
Obviously, conditions (i) and (iii) imply condition (i/, (iv) and (v) by
taking g{x) = 1. It is well-known that the mixing rate (iii) is essentially
sharp, even in the case of a finite second moment. However the following
example is interesting: Let X\ have the density function p(x) = a(l +
|ж|3)-1 for χ £ Д1, where a"1 = /^(l + lxl3)"1^. Let g(x) = exp(log(l +
|#|3)a) for some 0 < a < 1. It is easy to see that asx->oo
Н(х) ~2alog(l + H3)/3,
G(x) - 2a(log(l + \х\3))г-ад(х)/3а.
If p(^) < a/(51ogn), we can easily verify that the conditions in Remark
4.4.1 are satisfied but the condition (iii) in Theorem 4.4.1 fails. Hence
condition (iii) may be not essentially sharp in some particular case of infinite
variance, even of finite variance.
In order to prove Theorem 4.4.1, we introduce some notations. Let
M* be a positive integer such that
supH(x)/x2 > 1/M*. (4.4.1)
x>0

106
Chapter 4 Weak Convergence for p-mixing Sequences
For each η > M* define
tn = sup{x > 0 : H(x)/x2 > 1/n}. (4.4.2)
It is clear that tn —► oo monotonically as η —► oo. Note by a trivial
argument that
tl = nH(tn) for n>M*.
By condition (i), for any 0 < ε < 1/2 and large η
ηχ~ε < t2n < η1+ε. (4.4.3)
We need a few properties of these £n's.
Lemma 4.4.1. If condition (Hi) is satisfied, for any 0 < a < 1,
*fna]/*S—♦« asn-^oo. (4.4.4)
Proof. (4.4.2) implies
t2nH(t[na])/(tfna]H(tn)) —, 1/α as η - oo. (4.4.5)
We show that there is a M > 0 such that
limsup^/if <M. (4.4.6)
n—>oo
In fact, if not, there is a subsequence n^ such that limfc_>oo ^nk/^fn ά\ = °°·
Then using Property A5 of a slowly varying function (see Appendix) we
obtain
which is contrary to (4.4.5). By (4.4.2) and (4.4.6), it follows that
n/[na] < tl/t\na] < nH(Mt[na])/([na]H(t[na])),
which implies (4.4.4).
Lemma 4.4.2.
lim nP{\Xx\ >tn) = 0
η—►oo
and
lim {n/H{tn))ll2E\Xl\I{\X1\ > tn) = 0.

4.4 The WIP when the variance is infinite
107
The proof of this lemma can be found in Bradley (1988) and will be
not presented here.
Proof of Theorem 4.4.1.
For some ε > 0, put Cx = Cexp{(l + ε) ΣΕΤ1 P(?)}> where С is
defined in Lemma 2.2.2 and C2 = С exp{-(l + ε) ΣΕΤ1 p(20}> where c'
is defined in Lemma 2.2.3. For η > Μ* define
4n) = XkI(\Xk\ < tn) - EXkI(\Xk\ < in), к > 1 (4.4.7)
and
$) = ί) + ··; + 4η). m>l.
By the definitions of C\ and C2 we have
C2mEX[n)2 < E(S^)2 < СгтЕ(х[п))2. (4.4.8)
Put An(m) = \\Sm lb and An = An(n). Note that EX\ — 0, it is easy to
verify that
Е(Х[П))2 ~ H(tn) as n-^oo. (4.4.9)
Therefore there exist 0 < C'2 < C[ < oo such that
C'2nH{tn) <A2n< C[nH(tn). (4.4.10)
Next we formulate the proof in two steps.
Step 1. We prove that S^/An Л ЛГ(0,1) in distribution as η —► oo.
It suffices to show that
lim E{exp(itS^/An)} = exp(-*2/2) for any t G R. (4.4.11)
Since the case of t = 0 is trivial, it needs only to prove the above equality
for t φ 0. Fix t φ 0. Let J be a positive integer specified later on. Define
p* and L* to be positive integers such that
Kl-iViijOy-expi-i2/2)!^6/3 breach j>2L\ (4.4.13)
Let JV* > M* be a positive integer such that
N*>2p*-2L\ (4.4.14)
£(XJn))2 > 0 for each η > Ν*, (4.4.15)

108
Chapter 4 Weak Convergence for p-mixing Sequences
and
I28p*2t2
c2nEiXf^iEX'mu S W)
с2лс23/2 * n£(xjn))2 (n^(xin))2)3/2 J
<e/(3n) for each n>N*. (4.4.16)
Here, (4.4.16) can be justified by (4.4.9).
Let N > N* he an arbitrary but fixed integer. Then to prove (4.4.11)
it suffices to show that
\Eexp(itsP/AN) - exp(-£2/2)| < ε. (4.4.17)
Referring to (4.4.14), let L be the positive integer such that
p* < N/2L < 2p*.
Note that L > L*. Let ρ be the positive integer such that
p2L <N < (p+l)2L. (4.4.18)
It is easy to see that
p* <p<2p*. (4.4.19)
Let us partition N into disjoint blocks of consecutive integers, leaving
no gaps between the blocks. The order of the blocks is G(l), (5(2), · · ·, with
(p, if j is odd;
CardG(j) = < [2J+//2], where / is such integer that j/2l (4.4.20)
t is an odd integer if j is even.
Henceforth we shall deal only with the blocks G(l), G(2), · · ·, G(2L+1 - 1).
For each / = 1, · · ·, L, there are exactly 2L~l integers j £ {1,2, · · ·, 2L+1 —1}
such that j/2l is an odd integer. Therefore
L
Card{G(2) U G(4) U · · · U G(2L+1 - 2)} = ]T 2L~l[2J+lf2]. (4.4.21)
1=1
Hence by (4.4.18)
L L
N < 2Lp + Σ 2L_/ ^ 2Z> + Σ 2L_/[2J+//2]
1=1 1=1
= Card{G(l) U G(2) U · · · U G(2L+1 - 1)}
L
<7V + ^2L"/[2J+//2]. (4.4.22)
1=1

4.4 The WIP when the variance is infinite
109
For each j = 1,2, · · ·, 2L+1 - 1 define
uu) = Σ 4Ν)-
keG(j)
And further, for even integers ji and j'2 such that 0 < j\ < J2 < 2L+1,
define
V(jt,h) = U{jx + 1) + U{jx + 3) + · · · + U(h - 1).
If 1 < j < 2Z — 1, then for the integer m such that j/2m is an odd integer,
we have that m < I and hence (2/ + j)/2m is an odd integer, and hence
CardG?(2/ + j) = CardG(j) by (4.4.20). Consequently, if we denote и =
Card{i?(l) U G{2) U · · · U G(21)} and use the notation и + G = {u + g :
g £ G} for sets C? of positive integers, we have (by induction on j) that
G(2l + j) = и + (^(j) for j = 1,2, · · ·, 2l — 1. In particular, if we denote
G = G(l) U G(3) U · · · U G(2l - 1)
and
G* = G(2l + 1) U G(2l + 3) U · · · U G(2l+1 - 1),
then G* = u + G, У (2<, 2'+*) = Σ*€σ· 4*° = E*eG *£i and V(0,2') =
J]fcGGX^ \ The stationarity of the sequence \X\ \k > 1} implies the
following useful property:
For each / = 1, · · ·, L, V(0,2Z) and L(2Z, 2/+1) have the same distribution.
(4.4.23)
Hence by a simple calculation for each I = 1, · · ·, L,
2(1 - p([2J+l/2]))EV{0,21)2
<EV(0,2l+1)2
< 2(1 + p{[2J+ll2]))EV{<d, 21)2. (4.4.24)
Moreover for each I = 1, · · ·, L
\Eexp{itV(0,2l+1)} - (Eexp{itV(0,2l)})2\
< p([2J+l/2])E\exp{itV(0,21)} - 1|2
<p([2J+l/2])t2EV(0,21)2
< p([2J+ll2})t2Cx2l-lEU{l)2. (4.4.25)

110
Chapter 4 Weak Convergence for p-mixing Sequences
In what follows, it should be kept in mind that E(Sm ^)2 > 0 for all
m > 1 by (4.4.15), the fact N > N* and (4.4.8). By (4.4.24) and induction
L
π
1=1
2^{Π(ΐ-ρ([27+//2]))}^ί/χ2
L
EV(0,2L+1)2 <2L{[J(1 + p{[2J+42]))}EUl
and hence
1 -ε/2 < ||F(0,2L+1)||2/(2L/2||E71||2) < 1 + ε/2 (4.4.26)
provided J is large enough. By (4.4.21) and (4.4.8)
E{U{2) + E7(4) + · · · + U(2L+1 - 2))2
Ci(E2L+J"//2)£(Xi(,l))2·
L
<
1=1
Also by (4.4.22), U(l) + U(2) + ■■■ + E7(2L+1 - 1) - SJJ0 is the sum of at
most [Ei=i2L+J~l/2] distinct X(kN),s, and hence
E(U(1) + U(2) + ■■■ + U(2L+1 - 1) - SJJ0)2
L
Σ
1=1
Consequently, using (4.4.8), (4.4.12) and (4.4.19)
>L+1\ C(N)
\\V(0,2b+1)-S^>\\2 .
< ||[/(1) + E7(2) + · · · + U(2L+1 - 1) - S^h
+ ||i/(2) + E7(4) + --- + E7(2L+1-2)||2
<2ci/2(^2L+J-i/2)1/2||xiAr)|
1=1
L
1/2,
<2С\'22^^-112) ' \\U(l)h/(C2p)^
1=1
<2L/2£||i/(l)||2/2. (4.4.27)

4.4 The WIP when the variance is infinite
111
We now come back to (4.4.17). By (4.4.26) and (4.4.27) we have
AN
2b/2||[/(l)||2
<
||У(0,2^)||2| |||F(0,2^)||2-||5^||2
2W\\U{l)h I I 2b/2||[/(l)||2
e ||ηθ,2^)-5^||2
- 2 + 2b/2||[/(l)||2
< ε. (4.4.28)
(4.4.28) and (4.4.27) together imply that (4.4.17) is equivalent to
D := |£exp{ziF(0,2L+1)/(2L/2||i/(l)||2} -exp(-i2/2)| < e. (4.4.29)
Obviously
D < |£exp{ziF(0,2L+1)/(2L/2||[/(l)||2)}
-(£exp{i(i/2L/2)E/(l)/||i/(l)||2})2t|
+ |(Sexp«i/2L/2)E/(l)/||[/(l)||2})2L - (1 - (l/2)i2/2L)2L|
+ |(l-(l/2)i2/2L)2b-exp(-i2/2)|
=: ei + e2 + e3.
Using (4.4.25) and the elementary inequality
771 771 771
ΙΠ уь ~ Π Zk\ - Σl^ ~ Zfcl'
k=l k=l k=l
(4.4.30)
where yi, · · ·, ym, ζχ · · ·, zm are complex numbers in the closed unit disc,
we have
(^ехр{гТУ(0,2/+1)})2 - (Sexp{i7V(0,2z)})
< 2L-//9([2J+//2])T2Ci2/~1£;i/(l)2
for any T. Hence by induction
|Sexp{iIV(0,2L+1)} - (Яехр{гТУ(0,2)})21
L
< 2LT2ClEU{l)2Y^p{[2J+ll2}).
1=1
2L-l + l

112
Chapter 4 Weak Convergence for p-mixing Sequences
Letting Τ = £/(2L/2||t/(l)||2) and keeping in mind that U(l) = V(0,2), we
have
ei<t2CxX;p([2J+l/2])<e/3 (4.4.31)
1=1
provided the constant J is large enough. In order to estimate β2, define
the event
Fk = {\xlN)\=max\X<N)\}tk=l,-,p.
Put s = i/(2L/2||t/(l)||2) for simplicity. By (4.4.8) and (4.4.18),
s2 < t2/{2LC2pE{x[N)f) < 2t2/{C2NE{x[N)f).
Now we need a fact that for any real numbers χ and r,
\x - r\2 A\x- r\3 < 4r2 + 8(x2 A \x\3).
Using this fact and (4.4.19), (4.4.16) and (4.4.18) we have
£(|SC/(l)|2A|St/(l)|3)
<±EI(Fk)(\sPXr\2A\spXiN^)
<Ρ4Ε(\8ΧΙΝ)\2Α\3Χ[Ν)\3)
<p4{^2(EXxI{\Xi\<tN))2
+ 8Е{(з2Х21(\Хг\ < tN)) A QsflX^IdX^ < tN))}}
tE^XlH\Xl\<tN) л 23/2\t\3\X1\3I(\X1\<tN)y
* C2NE(x[N))2 С23/2ЛГ3/2(£(Хг(л°)2)3/2 ''
C^JV^XJ*0)2
400p*Vv|f|3)£rX12/(|X1|<^) л \xx?I{\Xx\<tN) л
n~ л r»3/2 I jvrcvviw^ λγ3/2<έυ у^КгАЗ/г J
< e/(3W) < e/(3 · 2L).
С2ЛС23/2 l JV^X^)2 JV3/2^^*0)2)3/2
0 < e/(3 · 2*).
Hence noting (4.4.30) we obtain
e2 < 2L|£exp(is[/(l)) - (1 - -s2£t/(l)2)|
< 2LE{\sU{l)\2 A \sU{l)\3) < ε/3. (4.4.32)

4.4 The WIP when the variance is infinite
113
Here we use the inequality about a characteristic function (cf. p.331 in
Bradley 1988).
As for ез, by (4.4.13) and noting L > L*, it is clear that
e3 < ε/3. (4.4.33)
(4.4.31), (4.4.32) and (4.4.33) together imply (4.4.29). This completes the
proof of the CLT.
Step 2. Now we show that
Wn => W as η -> oo. (4.4.34)
By Lemma 4.4.2 and (4.4.10) we have
Jirr^ P{ sup \S[nt] - 5^/j| > eAn} = 0 for any ε > 0.
Hence, in order to prove the theorem it suffices (cf. Theorem 4.1 in Billings-
ley 1968) to show that
W*=>W as n-^oo (4.4.35)
any 0 < t < 1
where W*(t) = S^l/An. By Theorem 4.0.4, it is enough to prove that for
Ai{[nt])/Ai —► i, as n-4» (4.4.36)
{W*(t)2,n> 1} is uniformly integrable, (4.4.37)
and there exists a constant λ > 1 for any ε > 0 such that for all large η
P{ max \s\n)\ > XAn\ < ε/λ2. (4.4.38)
We prove that
Al([mt})/Al(m) — t (4.4.39)
as m —> oo uniformly in η > Μ*, which implies (4.4.36). At first consider
the case of t — 1/p where ρ > 2 is an integer. Let q — [m/p],
(i+i)g
Yi= Σ χ]η)' * = o,i,...,p-i,
j=iq+l
m
yP= Σ ^1η)·

114
Chapter 4 Weak Convergence for /э-mixing Sequences
Note that
АЦт) = pAl(g) + £ EYiYj + £Yp2.
For гф 3 and integer /с > 1, by the Minkowski inequality
№^1<2||У4||2||5<я)||2 + р(*:)||У4||2||^||2.
Hence by (4.4.8)
|^(m)/^(i) -p\ < (p+ l)2(\\sin)\\2/An(q) + p{k) + ||Ур||^/^(д))
< (ρ + l)2((k2 + р2)т~1'г + p{k)) (4.4.40)
for every m large enough. Choose к such that (p + l)2p(k) < ε/2. Then
for m large enough
\A2n(m)/A2n([m/p])-p\<£
uniformly in η > Μ*. Therefore as m —> oo uniformly in η > Μ*
Л£(т)/Л£([т/р])—р. (4.4.41)
If £ is a rational number, that is, t — q/p for some integers ρ and g
with q <p, then
l2/r ,1W „2/ Ч_ А1([тя/Р}) АК™Я)
Al([mq/p])/Al(m) =
An(mq) АЦт)
q/p = t (4.4.42)
as m —> oo uniformly in η > Μ* by (4.4.41).
If £ is a irrational number, then for any given 0 < ε < 1/2, take rational
number t\ > 0 such that
ε/4 < t - ίχ < ε/2.
By the Minkowski inequality
\An([mt]) - An([mti])\ < An([mt] - [mtx]). (4.4.43)
Let ρ = [m/([mt] - [mix])]. Then
-ε-1 < m/(mt — τηίχ + 1) — 1
< ρ < m/(mt — mix — 1) < 5ε-1

4.4 The WIP when the variance is infinite
115
for m > 20/ε. Similarly to the proof of (4.4.40),
{p-{p+l)2p{k))A2n{[mt}-[mtx\)
< A2n(m) + (p+ l)2Al([mt] - [mh])A2n{k)
+ (p + l)2E(x[n))2.
Take к such that p(k) < ε/24. Then
A2n([mt] - [mh])/A2n{m)
< 6p-\l + 3(p + l)\A2n{k) + E(x[n))2)/A2n(m))
< 13ε (4.4.44)
provided m is large enough. Combining (4.4.44) with (4.4.43) and (4.4.42)
yields (4.4.39). Hence (4.4.36) is proved.
Now turn to (4.4.37). We have proved in Step 1 that
S[ni\/A[nt]^N(0,l) asn-oo
for any 0 < t < 1. FVom (4.4.8) and (4.4.4) we have
EVcW q(H)\24-2
-^[nt] ~ ^[nt} ) A[nt]
Η
= лйМЕад*и] <|Xi| ^tn))
i=l
< C2C^E(X2I(t[nt] < \Xt\ < in))/S(*i2/(l*i| < i[nt]))
= c2c;\H{tn) - H(t[nt]))/H(t[nt])
—> 0, as η —> oo.
It follows that
^η([ηί])/^[ηί] —► 1, as η -^ oo.
Therefore sfcl/An([nt\) Л N(0,1) as η -^ oo. By a well-known
result on uniform integrability (e.g. cf. Theorem 5.4 of Billingsley 1968),
(S^/Andnt]))2 is uniformly integrable and so is (5Ц/ЛП)2 by (4.4.36).
Finally we prove (4.4.38).
Let ln = ехр{ЕЙп1 ρ(2ψ3}. Define
X\? = XJQXil < tn/ln) - EXiI{\Xi\ < tn/ln),
X\? = XJ(tn/ln < \Xi\ < tn) - EXiI(tn/ln < \Xi\ < in),
i=l . i=l

116 Chapter 4 Weak Convergence for p-mixing Sequences
Obviously.
p{iWn\s^\>exAn}
< P{ max |4")| > XAn} + p{ max |s£>| > ЬХАп}
ζ='·Ρι+Ρ2- (4.4.45)
Note that for any integer К > 0
[log n] [log n]
£ p4/5(2*)<tf + p(2*)2/15 £ ρ{2ψ\
г=1 i=l
Hence, when in -> oo as η -> oo,
[log n] [log n]
Σ^') = ο(Σ//3(ή).
i=l i=l
Therefore, from Lemma 2.2.5, (4.4.2) and Property A4, it follows that
E\S^\5/2 < θ{^4(Ε(Χ^)ψ4
[logfc]
+ кехр{с^Р4/Ч*)}Е\х{?\5/2}
г=1
<
c{{kH{tn/ln))b'4
[logfc]
+ fcexp{c Σ P4/4?)}(tn/ln)1/2H(tn/ln)}
t=l
< c{(fctf (in)//n)5/4 + knll4H{tnfl4/ln}.
Using Lemma 4.1.2 we have
Ε max IS^I5/2 < c{{nH{tn)fl4/ln}(\ognfl2.
l<k<n
Without loss of generality we assume that р(2г) > l/(ilog2i). Then,
recalling (4.4.10) we obtain
Ε max |S£?|5/2 < c{nH{tn)fl4 < αΑψ. (4.4.46)
l<fc<n
Whence there exists a λ > 1 such that for each η large enough
pi < ε/λ2. (4.4.47)

4.4 The WIP when the variance is infinite
117
We now estimate p2. Let
r\ = [n/ln], r2 = [n/Z*], r = η + r2,
rfi = [(™ ~ n)/H d2 = [ra/r],
ir+r\
Yi= Σ X$> i = 0,l,---,du
j=ir+l
(t+l)r
z*= Σ *J!m i = 0,l,...,d2,
j=tr+ri+l
Tt(l) = J2YJ and Ti(2) = J2Z^
j=0 j=0
y*= Σ (|X;|/(t»/J»<l*jl<*»)
- E\Xj\I(tn/ln < \Xj\ < in)), t = 0,1, · · · ,dx.
It is easy to see that
p{m^xj4n)|>5A^n}
■ < P{ max \Ti{\)\ > 2\An) + p{ max |T<(2)| > 2\An)
ir+r\ *
+ 4--.Σ I^I^W
— jf=2rH-l
+ 2ίηρ{χπΐ|Χ2|5|2η)|>^λΑη}
=: h + h + h + /4. (4.4.48)
By (4.4.2) and (4.4.10), there is a A0 > 0 such that
nElX^IitJln <\Xi\< in)
< nlnt~xH{tn) < X0An.
Hence for λ > 8Ao
/3 < P{ max \Y*\ > 1\aA (4.4.49)
4<ζ<αι 4 '

118 Chapter 4 Weak Convergence for p-mixing Sequences
and by Lemma 2.2.2 and (4.4.10)
h < 2/„ρ{Σ(|Χί|/(ίη/Ζ„ < \Xi\ < i„)
i=l
- E\Xi\I{tn/ln < \Xi\ < *„)) > -^\An)
<cln{\An)-2r2EXp{tn/ln < \Х{\ < tn)
< с (XAn)-2nl-lH{tn)
< с \~2l~x < ε/λ2 (4.4.50)
provided η is large enough.
In order to estimate /χ, let
σ_! = (Ω,0), Gk = a(Xi,l<i<n + kr),
к
U0(0) = 0, Ui(k) = YiE(Yj+i\Gj+i-1), fc = 0,1, ···,<*!,
i=i
T*(fc) - rfc(l) - U0(k).
Obviously
/x < P{ max |T*(i)| > АЛП| + p{ max |C70(t)l > >лЛ
=:1[г)+1[2). (4.4.51)
Noting that {Т*(г), (?i, г — 0,1, · · ·, d\} is a martingale and using the
maximal inequality of martingale, we have
l[1] < 4(ААП)-2ЕТ*^1)2/(|Т*(^)| > \An). (4.4.52)
We prove below that for every г, к and η, by induction on к
EUf(k) < C1kr1p(r2)2\og2(2k)EX2I(tn/ln < \XX\ < tn). (4.4.53)
If к — 1, from the definition of p-mixing
EUf{l) = E(Yi+1E(Yi+1\Gi)) < p(r2)||yi+1||2||£7(yi+1|G0||2.
Thus (4.4.53 ) is true for к — 1 by a version of (4.4.8). If к > 2, assume that
(4.4.53) holds for every integer less than k. Put k\ = [&/2],&2 = к — k\.

4.4 The WIP when the variance is infinite
119
Then
EU2(k) = EU2(h) + EU2+kl(k2) + 2EUi(kl)Ui+kl(k2)
<EUHki) + EU?+kl(k2)
к
+ 2р(ъ)\\Щкг)\\2\\ Σ, Υί+*\
j=ki+l
< Cx{kx log2(2fci) + k2 log2(2k2) + 2k\/2kl12 log(2fc2)}
• rlP(r2)2EX2I(tn/ln <\Xi\< t„)
< Οφηρ{τ2)2^2{2^ΕΧ2Ι{ίη/Ιη <\Χι\< tn)
by induction assumption. This proves (4.4.53). Prom it and Lemma 4.1.2,
we have
Ε max U^{i)<ZCldlrlp{r2)2\ogi{2dl)H{tn)
l<i<d\
<cA2np{r2)2\og\2ln)
[bgn]
<οΑ2ηΡ(τ2)2(γ:ρ(2ψη.
i=l
Also,
[bgn] [bgr2]
Σρ{2ψ*< Σ ρ(2ψ* + p(r2)2/*log(n/r2)
t=l г=1
[logr2] [log η]
< р(г2)~1/3 Σ p(2i) + Мг2)2/3 Σ pW?iz
г=1 г=1
from which it follows
[log n]
Σ Ρ&Ψ3 = 0(Р(Ы~1/3) as η - oo.
г=1
Therefore we obtain
£ max U2(i) < cA2np{r2)2l* (4.4.54)
and further
/J2) < ε/λ2 (4.4.55)
provided η is large enough.

120
Chapter 4 Weak Convergence for p-mixing Sequences
For /2, having analogue to (4.4.52) and (4.4.55), we can get t'hat for
large η
Ι2<ε/λ2 + 4(λΑη)~2ΕΤΐ(2)
< ε/λ2 + AC1{XAn)~2d2r2EXll{tn/ln <\Χχ\< tn)
< ε/λ2 + 46Ί(λΛηΓ2ης1# (ίη)
< ε/λ2 + ολ-2*-1 < 2ε/λ2. (4.4.56)
Now we can come back to /} .
£Т*(^)2/(|Т*(^)| > XAn)
< 4£Tj1(l)/(|T,1(l)| > ^XAJ + AEUHd,)
< 36(E(S%fl(\S%>\ > X^)
+ E(± X^)2 + EUM))
i=d,2r
<144(^(5("))2/(|5(")|>λ^)
+ E(S^)2I(\S^\>X^)
12
+ ЕТЦ2) + EU2{dx) + Е(£ 4">)2).
i=d,2r
By (4.4.10), (4.4.54) and a version of (4.4.8)
A?(ETl(2) + EUM) + E(± X}2">)2)
i=d,2r
< c{n-42r2 + p(r2)2/3) < c{l~x + pin1'2)2'3)
< ε/2000
for large n. Using the uniform integrability of {(5„ ')2/Α^,η > 1}, we find
that
A-2E(S^)2I(\S^\ > \An/12) < ε/2000
for each η > 1 provided λ is large enough. Moreover, by (4.4.46)
A~2E(SS)2I(\S^\ > XAJ12) < 4X-^2A-5/2E\S^2 < cX~V2.
Whence we obtain that there is a constant λχ such that for any λ > λχ
and large η
/χ(1) < ε/λ2. (4.4.57)

4.4 The WIP when the variance is infinite
121
(4.4.55) and (4.4.57) together yield
h < 2ε/λ2. (4.4.58)
Proceeding exactly as the proof of (4.4.58), we also have
h < 2ε/X2 (4.4.59)
for any large λ and n.
It follows from (4.4.45), (4.4.47), (4.4.48), (4.4.50), (4.4.56), (4.4.58),
(4.4.59) that (4.4.38) holds, as desired. This completes the proof of
Theorem 4.4.1.

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Chapter 5 Weak Convergence for
<£-mixing Sequences
The CLT of a (^-mixing sequence is one of the earlier results for the
dependent random variables. Ibragimov (1959) gave the following two
Propositions.
Proposition 5.0.1. Let {ХП5 ™ > 1} be a strictly stationary φ-
mixing sequence of random variables with EX\ = 0, EX2 < oo. If
oo
Σφ1ΐ2{η)<οο,
71 = 1
then
oo
σ2 = EXf + 2j2EXixJ
3=2
converges absolutely, and if the condition σ > 0 is added, then Sn/a>/n
converges in distribution to -ZV(0,1).
Proposition 5.0.2. Let {Χη,η > 1} be a strictly stationary φ-
mixing sequence of random variables with EX\ = 0,Ε\Χι\2+δ < oo for
some δ > 0 and σ^ = ES% —> oo. Then Sn/an converges in distribution to
JV(o,i).
Since then, the CLT and the WIP for a (^-mixing sequence have ever
been discussed by many authors. Ibragimov-Linnik and Iosifescu have
raised the following conjectures:
Conjecture 1 (Ibragimov and Linnik 1971). Let {Xn, η > 1} be
a strictly stationary <£>-mixing sequence with EX\ = 0, EX2 < oo. If
σ^ —> oo as η —> oo, then the CLT holds.
Conjecture 2 (Iosifescu 1977). Let {ХП5^ > 1} be as above. Then
Wn weakly converges to W, where Wn(t) = 5[ηί]/ση.

124
Chapter 5 Weak Convergence for ^-mixing Sequences
Since 1970's, some mathematicians have obtained many beneficial
results around these conjectures. Herrndorf (1983b) showed that there exists
a strictly stationary <£>-mixing sequence with σ\ —> oo,liminfn_,oo σ^/η =
0, conjecture 2 does not hold. Peligrad (1985) proved that these two
conjectures hold true under the additional assumption liminf σ^/η > 0. Thus,
η—►oo
we can reduce the study of the above conjectures to that of the variances
of the partial sums. In those two papers they have also given some
sufficient conditions for the CLT and the WIP of a (^-mixing sequence when
the moment of order 2 is finite. We shall introduce these in Section 5.1.
In Section 5.2, we will discuss the above two conjectures and a general
conjecture which was raised by Peligrad (1990).
5.1 The WIP when the moments of order 2 are finite
Since p(n) < 2<^1/2(n), we have the same conclusion as in Theorem
4.1.1 with Σ£=ι <^1/2(2n) < oo instead of Σ™=ι P(2n) < oo.
Herrndorf (1983) dropped the condition on the mixing rate.
Theorem 5.1.1. Let {Χη? η > 1} be α φ-mixing sequence with EXn =
0, EX% < oo satisfying
(i) σ\ = nh(n), where h(n) is slowly varying,
(it) limn_+oo P{maxi<i<n \Xi\ > εση} = 0 for any ε > 0,
(Hi) {£^(η)/σ^, πι > 0, η > 1} is uniformly integrable.
Then
Wn=>W.
Proof. We are going to verify the conditions in Theorem 4.0.4. By
the definition of (^-mixing,
r r
\р{(}Е{}-]1Р{Е{}\<г<р([п6])^0 asn-oo,
г=1 г=1
where Ει = {Wn{ti) — Wn(si) £ Bi},i = 1, · · · ,r are defined in Theorem
4.0.4, δ = miii2<i<r(si — U-\) > 0. And hence, {Wn,n > 1} has
asymptotically independent increments. Moreover, conditions (i) and (iii) imply
the uniform integrability of {W%(t),n > 1} for each t > 0. Obviously
EWn(t) = 0 and EW%(t) —> t as η —> oo by condition (i). In order to show
the tightness, we need the following lemma.

5.1 The WIP when the moments of order 2 are finite
125
Lemma 5.1.1. Let {Χη? η > 1} be a φ-mixing sequence. For any
given positive integer q and α > 0, ra > 0, r > g + 1, we have
(1 - cp(q) - max P{|Sm+r - Sm+j\ > a})
χ P\ max |5m+J· - 5m| > 3a }
4<7<r J
< P{|Sm+r - 5m| > a}
+ PUq - 1) max \Xm+j\ > a]. (5.1.1)
I l<j<r J
Proof. Denote A\ = {|Xm+i| > 3a},
Aj = {\Sm+j - Sm\ > 3a, |Sm+i - Sm\ < 3a, 1 < г < j - 1}, 2 < j < r,
#i = {|Sm+r - 5m+j+g_i| < a}, 1 < j < r - g,
Β,=Ω, r - q + 1 < j < г, С = {|Sm+r - 5m| > a}.
It is clear that
r
(J Λ,·Β,- С C{j{(q - 1) max |Xm+i| > a}.
^'-
Hence
P(C) + P{(q - 1) max \Xm+j\ > a}
3=1 3=1
> {i<mjn_ P(Bj) - ψ{ς)} J2 P(A3). (5.1.2)
i<ii=- , j=1
Note that
r
Y] P(Aj) = Pi max. \Sm+j - Sm\ > 3a),
i=i
min P(5j) + max P{|Sm+r - Sm+j\ > a} > 1.
l<j<r—q 4<3<τ
Inserting these into (5.1.2) implies (5.1.1), as desired.
In order to prove that {Wn} is tight, it needs only to show that
lim - max limsupPJ sup |Wn(s) — Wn{kS)\ > ε\ = 0.
$10,1/6GN О 0<к<1/6 n-^oo 1к6<з<(к+1)6 J
(5.1.3)

126
Chapter 5 Weak Convergence for ^-mixing Sequences
Choose a positive integer q so large that φ{α) < 1. For any given ε > 0
and δ > 0, by condition (iii) we have
sup sup P{\Sm(j)\ >εση/3}
m>0l<j<n6
<9e~2 sup E(Sl(j)/a]) sup σ]/σ2η
m>QJ>l l<j<n6
= C(e) sup σ]/σ2η.
1<3<ηδ
It follows from (i) and Property A4, that
limlimsup sup σ· /σ^ = 0.
<H0 n->oo \<j<n8
Therefore, there exists a 6q = δο(ε) > 0 such that for any 0 < δ < δο(ε)
limsupsup sup P{|Sm(j)| > -εση\
n->oo m>0 1<j<n6 L О J
<C{e)6<±(l-V{q)). (5.1.4)
Applying Lemma 5.1.1 for m = [пкб], r = [n{k+ 1)δ] — [пкб], За = εση,
from (5.1.4) we obtain that for any 0 < δ < δ0 with Ι/δ G Ν, 0 < к < 1/δ,
J(l - ^limsupPJ sup \Wn(s) - Wn(k6)\ > ε}
l n->oo ^k6<s<(k+l)6 J
< HmsupP{|Wn((A; + 1)δ) - Wn(kS)\ > ε/3}
+ limsupP\ (q — 1) max \Xj\ > -εση}
ΤΪ.—►ОТ) ^ 1<7<П J 3
n->oo v 1<7<™
= :Ii+h (5.1.5)
For fixed g, it follows from (ii) that 1<ι = 0. And from (i)
^σΗ^ι)δ]-^δ]/ση = <51/2·
Thus we have
h < HmsupP{|5[n(fc+1)(5] - 5[nfc(5]| > εσ^+^^^/Α^/δ}
п—юо
<16ε~2£ sup ESl(n)I(\Sm(n)\>ean/(4V6))/al
m>0,n>l
Combining it with (5.1.5) and condition (iii) implies (5.1.3). The proof of
Theorem 5.1.1 is completed.

5.1 The WIP when the moments of order 2 are finite
127
Corollary 5.1.1. Let {Xn η > 1} be a φ-mixing sequence with EXn =
0, EX2 < oo. If the conditions (i) and (ii) of Theorem 5.1.1 and
(iv) sup{£;5^(n)/a^, m > 0, η > 1} < oo
are satisfied, and if {Xnn > 1} obey the CLT, then Wn => W.
Proof. From the CLT and (i), it follows that S\nt^/an converges in
distribution to W{t) for every t > 0. Now let 0 < 5 < t be given. We shall
show
(S[nt]-S[ns])/an^W(t)-W(s) asn-^oo. (5.1.6)
It is obvious that {(S[ns]/an, S[nt]/an), η > 1} is tight (see Billingsley 1968
p.41). Hence, it follows from Helly's theorem that there exists a probability
measure Q on R2 and a subsequence {n^} such that
(s[nks]/<7nk, S[nkt]/°nk) -+Q as A: -^ oo.
Let 7T{ : R2 —> ϋ, г = 1,2, denote the projections. Then
(s[nk8]/<rnk, (s[nkt) - s[nks])/<7nk) -► Q{*u π2 - πι)-1 as к -► oo.
Taking ρ = p(n) G {0,1,..., [ns]} such that p(n) —> oo and ρ(η)/ση —> 0
as η —* oo, we have
(ESfns]_p{n^(n))Y/2/an
< ρ{η)σ~ι sup(£X?)1/2 -> 0 as η -^ oo.
3
Hence it follows that
(S[nks]-v{nk)/°nki{S[nkt] - S[nks})/<7nk)
—> Q(tti, π2 — ΤΓχ)-1 as к —> oo.
For any Borel sets A,B С R with £?(πι, π2 — /τγι)~1(^(^4 χ В)) = 0, we
obtain by an application of the mixing condition
|<3(πι, π2 - i^)'\A x В) - Qn-\A)Q(n2 - πχ)"1^)!
= ^ί^Ι^ί^βΙ-ρίη*)/^* € Л, (5[njfet] - S[nk8])/ank G β}
- ^{5[nfc5]-p(nfc)/°W ^ ^^{(^[п**] - s[nk8])/°nk £ £}|
= 0.
Hence πχ and π2 — πχ are Q-independent. Since Q7rf1 = N(0,s) and
Qtt"1 = JV(0,i), it follows that Q(tt2 - πχ)-1 = JV(0,i - s). This proves
(5.1.6).

128
Chapter 5 Weak Convergence for ^-mixing Sequences
The remainder of the proof needs only to check (5.Γ.3). From (5.1.6)
it follows that
limsup P{\Wn((k + 1)6) -Wn{k6)\ >ε/3}<Ν(0,δ){χ: \x\ > ε/3}.
η—>oo
(5.1.7)
Furthermore
1
cN(0,6){x: \x\ >ε/3}
о
^2 3ν^ 1 _£2/18^о ^ δ^ο
δ ε уДк
Now (5.1.5) and (5.1.7) imply (5.1.3). This completes the proof of
Corollary 5.1.1.
Remark 5.1.1. The conditions (i) and (ii) of Theorem 5.1.1 are also
necessary for the WIP.
Suppose that Wn weakly converges to W. For δ > 0 and a function /
let w(f,6) = sup{\f(x)-f(y)\ : 0 < x, у < 1, \x-y\ < δ}. For any ε > 0,
the set {/ : / G D[0,1], w(f,6) > ε} is closed with respect to the uniform
topology. Hence
HmsupP{ max \X{\ > εση} < \im sup Ρ{w(Wn, δ) > ε}
п—юо 1<г<п η—юо
< P{w(W, δ)>ε}-+0 as<5 -> 0.
This proves that (ii) holds true.
In order to prove (i), we show that h(t) := max(l,am)/£,£ > 0, is
slowly varying. Since σ\ —> oo as η —> οο,/ι(£) = <rftJt for large t. For
t G [0,1] we have
S[nt]/<T[nt] -+ N(0,1), s[nt]/^n -► iV(0, i) as n -^ oo.
Therefore we obtain for every t G [0,1]
σ\ηΐ\ΙσΙ—►* as n-^oo. (5.1.8)
ρ
Moreover (ii) implies Χη/ση —► 0. Hence both Sn/an and Sn/an+i weakly
converge to -ZV(0,1) and further ση/ση+χ —> 1. Consequently for £ G [0,1]
σ[*β]/σ[*[β]] —► 1 as s -► oo. (5.1.9)

5.1 The WIP when the moments of order 2 are finite
129
From (5.1.8) and (5.1.9), we obtain lims_>oo h{ts)/h{s) = 1 for every t £
[0,1]. Hence h is slowly varying.
Now we can write the following Corollary for the strictly stationary
case immediately.
Corollary 5.1.2. Let {Xn,n > 1} be a strictly stationary φ-mixing
sequence with EX\ = 0, EX\ < oo and σ\ —> oo. Denote
Yn(t) = —(5[nt] + (nt - [nt])X[nt]+l).
Then the following assertions are equivalent
(a) Wn=*W,
(b) Yn^W,
(c) {Xn} obeys the CLT and (ii) is fulfilled,
(d) {5^/σ^,η > 1} is uniformly integrable and (ii) is fulfilled.
Proof. Obviously, (a) and (b) are equivalent and (b) implies (c).
From Theorem 5.4 of Billingsley (1968), it follows that (c) implies (d).
Finally "(d) implies (a)" follows from Theorem 5.1.1 .
Peligrad (1985) weakened the conditions of Herrndorf (1983) and gave
the following theorem.
Theorem 5.1.2. Let {Xn,n > 1} be a φ-mixing sequence with EXn =
0,EX% < oo. Suppose that the conditions (i), (iv) and the Lindeberg
condition
(v) limn^oo 4r ΣΓ=ι EX?I(X? > εσ2η) = 0 for any e > 0 are
satisfied. Then Wn => W.
Remark 5.1.2. If φ* = Нтп_юо φ{η) < 1, then (ii) is equivalent to
η
J2P{\Xi\ > ^n} —► 0 for any ε > 0 (5.1.10)
г=1
and the Lindeberg condition (v) is equivalent to
EmaxXf/oi^O. (5.1.11)
1<г<п
In fact, (5.1.10) implies (ii) obviously. On the other hand, by (ii) we can
choose no and po such that
P{ max X? < εσ2Λ - φ(ρ0) > а > 0 for η > n0. (5.1.12)

130
Chapter 5 Weak Convergence for ^-mixing Sequences
Therefore for every χ > ε > 0, η > no and j = 0,1, · · · ,ρο — 1 we have
p{mgnX?>*o£\
>ρ{χ]>χσ2η, max Xf < χσ2Λ
{ Po+j<i<n J
+ ■·■ + P{X[(n-j)/p0}po+j ^ Χση)
> Σ пЧо* * *<*}
0<i<[(n-j)/p0]
(Pfen^<X<T«}-^0))
0<i<[(n-i)/po]
which implies
Σ P{X? > ^} < ^P{maxnx? > x^}. (5.1.13)
Hence we can obtain (5.1.10) from (ii). The fact that (v) implies (5.1.11)
follows from
Ε max XfΊσ2
1<г<п г ' П
< ε + Ε max Xfl(Xf > εσ2η)/σ2η for any ε > 0.
1<г<п
Note that, (5.1.12) and further (5.1.13) hold under (5.1.11). Hence, (5.1.11)
implies that (v) follows from the following well-known relation: for every
positive integrable random variable X,
EXI(X >b) = bP(X >b)+ P(X > x)dx. (5.1.14)
Remark 5.1.3. Utev (1990) showed that for a (^-mixing sequence
{Xn}, the Lindeberg condition (v) implies the CLT. Furthermore, Grin
(1991) showed that for a stationary (^-mixing sequence {Xn} with φ(1) < 1,
there exists a sequence {An} of numbers such that B~1Sn — Лп —► -ZV(0,1),
where Bn = sup{z > 0 : Var(Zi=iXJ(\xi\ < *)) > r2}, if and only if
{Bn} is regularly varying with the exponent 1/2.
The proof of theorem 5.1.2 needs the following lemmas. First we state
an analogue of Lemma 2.2.7.

5.1 The WIP when the moments of order 2 are finite
131
Lemma 5.1.2. Let {Yn->n > 1} be a sequence of random variables.
Denote Tn = Σ?=ι Yi- 4 for some b > °>P G N and ao > 0
φ(ρ) + max P{Tn - T{\ > ba0/2} < η < 1, (5.1.15)
1<г<п
then for every a > ao and n > ρ the following relations hold:
p{ max \Т{\ > (1 + b)a\ < P(\Tn\ > a)
^1<г<п У 1 — Г\
+ —ί-Pf max \Yi\ > . Ьа А(5.1.16)
Ι-η li<i<n' ' 2(i? - 1) J
and
P{\Tn\ > (1 + 2b)a} < JL-P{\Tn\ > a}
1-η
+ -ί-Ρ( max \Yi\ > — }. (5.1.17)
For simplicity, denote
EaX = EXI(X > a).
Lemma 5.1.3. Let {Yn,n > 1} be a sequence of random variables
satisfying (5.1.15). Then for every A > a$ we have
Е(1+2ЫаТ% <(1 + 2bf^EATl
/2p(l + 2b)\2 1 π „
Proof. By (5.1.14) and a change of variables one obtains
?(l+26)2
#(i+26)MTn2 =(1 + 2bfAP{Tl > (1 + 2b)2A}
+ (l + 2b)2/ P{Tn2>(l + 2b)2y}dy.
The lemma follows by (5.1.17) and then (5.1.14) again.

132
Chapter 5 Weak Convergence for ^-mixing Sequences
Lemma 5.1.4. Let {Xn,n > 1} be a centered sequence such that φ* <
1/4 and {maxi<2<n£'X?/a^,n > 1} is bounded. Then
\m^E{Sn-Si)2/a2n,n>l\
is bounded as well.
Proof. Let ρ be an integer such that φ(ρ) < 1/4. We have
max E(Sn - Si)2 < max E(Sn - Si)2 + p2 max EX2. (5.1.18)
1<г<п 1<г<п—р 1<г<п
For every г < η — ρ we also have
|||S„||2 - \\Si + (S„ - 5i+p)||2| < ρ max ||Хг-||2.
1<г<п
By Lemma 1.2.8, we have
> (1 - 2¥?1/2(ί,))1/2(σ2 + E{Sn _ Si+pfYl\
Therefore
d < (1 - 2^1/2(ρ))"1/2(ση +p max \\Х{\\2)
\ 1<ι<η /
for every г < η — p. Whence, from (5.1.18)
max E(Sn - Si)2/a2n
1<г<п
< 2 + 4(1 -2φιΙ2{ρ))-1
+ p2(l + 4(1 - 2//2(ρ)Γ1) max EXf/a2n
1<ι<η
as desired.
Lemma 5.1.5. Let {Xn,n > 1} be a centered sequence with ф* < 1/4.
Then {maxi<2<n S2 /σ^η > 1} is uniformly integrable if and only if
{maxi<i<nX?/a^,n > 1} is uniformly integrable.
proof. First, because
P{ max \ХЛ > 2χσΛ < p{ max \SA > χση\ (5.1.19)
U<t<n ' J "" ll<z<n ' J V J
for any χ > 0, one of the implications follows by the relation (5.1.14).

5.1 The WIP when the moments of order 2 are finite
133
We prove the part of "г/", and hence, assume that {maxi<2<nX^/a^,
η > 1} is uniformly integrable. By the Chebyshev inequality and by
Lemma 5.1.4 for any b > 0
lim sup max P{(Sn - Stf > {Ъ/2)2Ьа2п} = 0. (5.1.20)
t—>oo n 1<г<п
By φ* < 1/4 and (5.1.20), we can find some constants Ь > 0,77 < 1/2,ρ G N
and ao G R such that
(1 + 26)V(1-4)<1 (5.1.21)
and
φ(ρ) + max P{(Sn - Si)2 >(b/2)2a2a2n} < η
1<г<п
for everyn > 1. (5.1.22)
Prom Lemma 5.1.3 and (5.1.21), (5.1.22) it follows that
+
/2p(l + 2b)\2 1 / Х?ч
(—ь ) i^^Hm^)
for any Л > Oq and every η > 1. Taking the supremum on n in this
relation and noting that supn ^(5^/σ^) is decreasing in A and that
{maxi<i<n(X^/a^),n > 1} is uniformly integrable, we obtain
lim 8ърЕА(Щ) < (1 + 2Ь)2-^- lim 8ирЕА(Щ).
Whence it follows by (5.1.21), (5.1.22) that {БЦа1, п > 1} is uniformly
integrable. This implies by (5.1.16) and (5.1.14) that {maxi<;<n Sf/σ^π >
1} is uniformly integrable.
Proof of Theorem 5.1.2. By the proof of Theorem 5.1.1, {Wn,n >
1} also has asymptotically independent increments. By Remark 5.1.2 it
follows that {maxi<i<n(X?/a^),n > 1} is uniformly integrable under the
Lindeberg condition. Whence {W2{t),n > 1} is uniformly integrable for
each t by (i) and Lemma 5.1.5. Moreover, EWn(t) = 0 and EW2{t) —► t
as n —> 00 by (i) again.
For the tightness condition of {Wn, n > 1}, from the proof of Theorem
8.3 of Billingsley (1968), it suffices to show
[1/«Ы .
limlimsup V PJ max \Wn(s)-Wn(i6)\ > ε\ = 0. (5.1.23)
δ—>0 η—>οο Τ~ί 46<«<(г+1)6 J

134
Chapter 5 Weak Convergence for (^-mixing Sequences
For every 0 < г < l/<5 — 1 denote
(ί+1)ηδ ,
Si = fi(n,6,a) := max p{\ V Xk\ > -α1/2ση\.
in6<j<(i+l)n6 *■ I 7^7. I Z ^
/c—j
By the Chebyshev inequality we have
9 9 1 (»'+l)n* 9
/. < (Λ2Ι max #( у χΛ2/σ2.
K—J
By (i), (iv) and the properties of a slowly varying function that follows
from the Karamata representation (see Appendix) we obtain
limlimsup max f% = 0. (5.1.24)
6-+0 n-+oo 0<t<l/$-l
Choose ρ and b such that
V(p)(l + 2b)2/(l-v(p))<l
and choose δο and no such that for any δ < δο and η > щ
φ{ρ) + max /; = η'(η, δ, α) := η1 < 1. (5.1.25)
1<г<1/6
Prom (5.1.25) and Lemma 5.1.3 we obtain for every 0 < г < 1/δ — 1
E(l+2b)*a(( Σ Xi) /ση)
ni8<j<n{i-\-l)6
< (1 + 2b)2_gLg/g»«^<ntf+l)f^·^
/2р(1 + 2Ь)ч2 1 ,Χ?.
+ 1 ь J ГГ^ L· E°W2p)'[-jr)· (5.1.26)
Noting conditions (i) and (iv), for fixed δ > 0 we have
u£f»p Σ E( Σ *i) /ση
г=0 ni8<j<n(i+l)8
= 0(limsup Σ σ^,/σ*) = O(l).
Denote
П^°° i<l/6
[1/4-1 2
/(a) = limsuplimsup Σ i£a( ^ Xyj /σ^.
$-+0 n-^oo .=0 nt6<j<n(t+l)6

5.1 The WIP when the moments of order 2 are finite
135
From (5.1.26), (5.1.24) and condition (v) we obtain
/((1 + 2b)2a) < (1 + 26)X(p)/(a) for everya > 0.
1 - φ[ρ)
Since 1(a) is a decreasing function in a, and [(1 + 2b)2(p(p)]/(l — φ(ρ)) < 1,
we obtain lima_>o 1(a) — 0· Hence 1(a) = 0 for every a > 0, which implies
[l/8]~l
limsuplimsup ]jP P(\ ]jP ΧΑ > εση) = 0
δ-+0 n-+oo .=0 n«<i<n(i+l)6
for any ε > 0. The relation (5.1.23) follows now by (5.1.16), (v) and
(5.1.24). The proof of Theorem 5.1.2 is completed.
Particularly, by Theorem 5.1.2 we have the following corollaries.
Corollary 5.1.3. Let {Xn,n > 1} be a stationary φ-mixing sequence
with EX\ = 0,EX2 < οο,σ^ —> oo and the Lindeberg condition (v) is
satisfied. Then Wn => W.
Corollary 5.1.4. Let {Xn,n > 1} be a strictly stationary φ-mixing
sequence with EX\ = 0, EX2 < oo, σ\ —> oo and for any ε > 0
lim ^EX\l(X\ > εση) = 0. (5.1.27)
Then Wn => W.
A special case of Corollary 5.1.4 is
Corollary 5.1.5. Let {Xn,n > 1} be a strictly stationary φ-mixing
sequence with EX\ = 0,EX2 < oo and liminfna^/n > 0. Then Wn =>>
W.
Remark 5.1.4. Peligrad (1985) pointed out that in some cases the
Lindeberg condition (v) also is necessary: If {Xn,n > 1} is a (^-mixing
sequence such that Wn => W, σ\ —> oo and φ(ΐ) < 1, then the Lindeberg
condition (v) is satisfied.
In fact, by Remark 5.1.1 we have σ2 = ih(i), where h is a slowly varying
function on i?"1", whence,
{ max E(Sn - Б{)2/а2п,п > l)

136
Chapter 5 Weak Convergence for yi-mixing Sequences
is bounded. So there exists a ίο > 0 such that for every η
φ(1) + max P{\Sn - 5<| > t0an} < с < 1.
1<г<п
Then by the proof of Lemma 2.2.7 for any χ > t^ and for each η G N, we
have
P{ max S? > 4хаП < -J-P{S£ > χσ2η}. (5.1.28)
4<г<п J 1 — С
On the other hand, the weak convergence to W implies the uniform inte-
grability of {Sl/σΙ,η > 1}. This fact together with (5.1.28) and (5.1.14)
implies {maxi<i<n Sf/σ^,η > 1} is uniformly integrable. By Lemma 5.1.5
{тахккп Xf/a„, η > 1} is also uniformly integrable. From Remark 5.1.1
we have
lim P\ max \XA > εση > = 0 for any ε > 0.
η—>oo ll<i<n J
Hence we obtain (5.1.11) since {max1<i<nX?/a^,n > 1} is uniformly
integrable. By Remark 5.1.2, (v) is a necessary condition for the weak
convergence to W.
5.2 The Ibragimov-Linnik-Iosifescu conjecture
We have showed in Corollary 5.1.5 that the Iosifescu conjecture is true
under the assumption liminf σ^/η > 0. But Herrndorf (1983b) showed by
an example that the Iosifescu conjecture does not hold true if lim inf σ^/η =
0.
Example 5.2.1. Assume that {ηη,η > 1} be a strictly stationary
(^-mixing sequence with Εηη = 0, Εη„ < oo, τ^ — Ε\Υ^=1ηΛ —> oo,
lim inf T^/n — 0. If {ηη, η > 1} does not satisfies the WIP, then we can
take Xn = ηη for η G N. Suppose that {ηη,η > 1} satisfies the WIP. Let
{an,n > 0} be a sequence of independent identically distributed random
variables, which are independent of {ηη} and satisfy
P{a0 = актПк} = bkn^1 fork G N,
Ρ{αο = 0} = 1-Σ6*η^ (5*2Л)
к
where the positive integers щ < ri2 < п$ < · · ·, and the sequences of real
numbers {а&}, {Ь&} with α*. —> oo, Ъ^ —> oo such that
Σ 6*»fcx ^ V2, Σ °*<6*η*1 < °°- (5·2·2)
fc A:

5.2 The Ibragimov-Linnik-Iosifescu conjucture
137
( For instance, from τ%/η —> 0, we can choose пь such that т%кпк г < 2 2k l
and rik to be increasing, a^ = /с, b^ = 2k.) (5.2.1) and (5.2.2) imply
Eal < oo, P(a0 = 0) > 1/2. Now, define
Χη^Ήη + Οίη- αη_χ. (5.2.3)
Denote Sn = Σ]=ι Xj, Tn = Σ?=1 Vj- We have
5n - Tn + αη - α0. (5.2.4)
It is clear that {Xn,n > 1} is a strictly stationary (^-mixing sequence with
EXi =0,EXl <oo, and
°l/T2n - 1. (5.2.5)
For n, m > 1 we can write
/Tw+m - Tmx 2 /^2 + /qn+m - qmx 2
It is well-known that if {ηη} satisfies the WIP, then {(Tn+m—Tm)2/r^, m >
0,n > 1} is uniformly integrable. Thus, from (5.2.6), (5.2.5) and
||(an+m - am)/an||2 < 2||α0||2/ση -► 0, (5.2.7)
it follows that {(5n+m — 5m)2/a2, m > 0,n > 1} is also uniformly
integrable. Since {ηη} satisfies the WIP and the random vector
σΰl (S[nt!] J ' ' ' J S[ntk] )~ТП1 (T[ntl] J ' ' ' J T[ntfc] ) -^ 0
for any 0 < ti < ··· < tk < 1, (W[nti]5"-5W[ntfc]) converges weakly to
W^"1..^, where Wn(i) = 5[ηί]/ση. We shall show that Wn does not weakly
converge to W. For t>0we have
P\ max |5i| > £ση \
ll<t<n ' "" J
> p{ max |ai - a0\ > 2tan\ - p{ max \T{\ > tan\, (5.2.8)
P<{ max \ai — αο| > 2£ση >
^ 1<г<тг J
> P{ao = 0, max \ai\ > 2tan \
l<i<n J
> -P{max \a{\ >2tan\
2 l<i<n J
= \(1 - (Ρ{|αο| < 2ίσ„}Γ)
> i - i exp(-nP{|a0| > 2ίσ„}). (5.2.9)

138
Chapter 5 Weak Convergence for (^-mixing Sequences
Using (5.2.9), (5.2.5), α& —> oo, (5.2.1) and bk —> oo, we obtain
limsupP<{ max \a.i — ao\ > 2tan \
- о " о ,lim exp(-nfcbfc/nfc) = -.
Since {77η} fulfills the WIP and (5.2.5), we can choose to > 0 with
HmsupP{ max \T{\ > toan} < 1/4.
n—юо 1<г<тг
Then (5.2.8) and (5.2.9) imply:
HmsupPJ max \Si\ > tan\ > - for every t > to?
whence {Xn5^ > 1} does not satisfy the WIP.
Furthermore, Peligrad (1990) pointed out that conjectures 1 and 2 are
not the most general results which one can expect for a (^-mixing sequence.
After comprehensive survey of the relevant works, she showed that the
following conjecture might be true.
Conjecture 3 (Peligrad 1990). Let {Xn,n > 1} be a strictly
stationary centered (^-mixing sequence satisfying:
H(x) := ΕΧ^Ι(\Χλ\ < χ) is slowly varying as χ —► oo (5.2.10)
and φ(1) < 1. Then Wn weakly converges to W, where
wn(t) = s^/dn^y^bn) о < t < i,
bn = E\Sn\.
Remark 5.2.1. There are at least two situations of interest when
it is easy to verify (5.2.10). One is EX\ < oo, and the other is that
P(|Xi| > x) is regularly varying with the exponent —2, i.e.,
Р(\Хг\ > χ) = l/(x2h(x)), (5.2.11)
where h(x) is slowly varying as χ —► oo.
In fact, we can rewrite Ρ(|Χχ| > χ) — h(x)/x2. By the partial
integration we have
H(x) = - jXy2dP{\X1\>y)
Jo
= -h(x) + [ h(y)y~1dy.
Jo

5.2 The Ibragimov-Linnik-Iosifescu conjucture
139
By the Karamata representation (see Theorem Al), we can choose z\ —
z\{x) < χ such that
limsup sup h(y)/h(x) — 1,
x—>oo z\<y<x
lim x/zAx) — oo.
x—>oo
So that
PX ГХ |
/ h{y)y~xdy> I h(y)y~1dy> -h(x)\og(xz^1).
J0 Jzi *
Then
Я(х) = (1+о(1))Г%)у-^.
Jo
For any given fc>0 we have
ρ kx px
| у h(y)y-1dy\ < 2(log fc)M*) - o(j h(y)y-1dy).
It follows that
lim Hikx)/H(x) = 1.
x—>oo
Peligrad (1990) proved that Conjecture 3 is true under condition (5.2.11).
Theorem 5.2.1. Let {Xn,n > 1} be a centered, strictly stationary
φ-mixing sequence satisfying (5.2.11) and φ(1) < 1. Then
Wn=>W.
The proof of Theorem 5.2.1 will not be presented here.

Chapter 6 Weak Convergence for
Mixing Random Fields
There are two kinds of definitions of mixing dependence for a
random field. One is a natural generalization from the classical case, the
sequence of dependent random variables, to the dependent random field,
which has been discussed by Bulinskii-Zurbenko (1981), Gorodezkii (1982,
1984), Bolthausen (1982), Nahapetian (1987), Bradley (1992), Donkhan
and Guyon (1991), Guyon(1992) and Donkhan (1994), etc. Another is
appeared in the study of set-index partial sum process for weakly dependent
random fields, which has been discussed by Goldie and Greenwood (1986
a,b), Chen (1991) and Lu (1995), etc. We shall introduce the first case in
Section 6.1 and the second case in Sections 6.2 and 6.3 respectively.
6.1 The CLT for mixing random fields
A natural generalization from an α-mixing sequence to an α-mixing
random field has been discussed by some mathematicians.
A random field {£t?t € %d}id > 1, is said to be a*-mixing, if
(r) = sup{|P(AB) - P(A)P(B)\ : A G συ, Β G ay,
U,V CZd,\U\ <m,\V\ <n,d(U,V)>r}
-► 0 as r -► oo. (6.1.1)
where d(U,V) = inf{d(t,s) : t G E/, s G V}, rf(t,s) = maxi<;<d \t{ —
Si|, m,nGNU {°°}5 σΑ — cri&jt £ ^4}> |-<4| is the number of elements of
A.

142
Chapter 6 Weak Convergence for Mixing Random Fields
For some subsets Δ^· ,j — 1, · · ·, fc, of d-dimensional cube Jn — [—n, n]d,
denote
5(n, j) - σ~χ5 (η), σ2 = VarSJn,
i
S/ = E&> *czd, |/| < oo,
к к
Mk = \E]\ eits(n^ - JJ EJtSW>l
3=1 3=1
к
MM) = Σ / №>i)№). ε, 5 > 0. (6.1.2)
Nahapetian (1987) proved the following conclusion, which is a
generalization of Theorem 3.2.1.
Theorem 6.1.1. Let {Xt?t £ ^d} be α strictly stationary a*-mixing
random field with EXt = 0,E\Xt\2+s < oo /or some δ > 0. ///or some
r >0,
(%J <2m,n(r) < f(m)riTa(r), where f(m) is a non-negative function,
m G NU{oo};
(it) E^i^-V/(2+*)(r) < oo, a(r) = o(r-(2r+1)d), r -^ oo,
Йеп
σ2 - ]Γ £X0*t < oo,
tezd
and г/ σ φ О, we /mve
sJnK4jv(o,i). (6.1.3)
The proof of Theorem 6.1.1 will need some lemmas. It is clear from
the proof of Lemma 1.2.3 that we have
Lemma 6.1.1. Let random variables X and Υ be measurable with
respect to σ-fields συ and σγ respectively, \U\ < ra, |V| < n, d(t/, V) >
r, E\X\p < oo, E\Y\q < oo, p, q > 1, p~l + g"1 < 1. Then
\EXY - EXEY\ < сЩХ^РЩУ^^а^г'1-*'1^).
Particularly, if \X\ < C\ a.s., \Y\ < C2 a.s., we have
\EXY - EXEY\ < cCiC2am,n(r).

6.1 The CLT for mixing random fields
143
Lemma 6.1.2. Let £i, · · ·, ζη be a sequence of random vectors, \E П^=г
x£j| < oo, г = 1,··· ,n — 1, |£7&| < 1, г = 1,·· ·,τι. Then
\Eiib-iiEb
s=l »=1
< Σ Σ Hte - !)te -!)
г=1 j=z+l
χ Π Ь-Е&-1)Е&-1) П 6
s=7 + l s=j+l
Proof. Obviously, we have
*i j-j, 77, 1 77, 77,
ΗΠ^-Π^|^Σ|^ Π б-я&я Π 6·
s=l s=l г=1 jf=z+l j=i+l
|я&
η
Π о-
j=i+l
< \e& -
-Efo
ΕξιΕ Π
l)(6+i - ]
- 1)Я(6+1
6
ο Π ο
η
.-ι) Π
6
j=i+2
(6.1.4)
(6.1.5)
+ |я(й-1)й+2 п е,--ад-1)^б+2 Π fc
i=i+3
By this recursive process, we obtain
j=i+Z
Eti Π ti-EiiE Π 6
i=»+i j=i+i
< Σ |я(&-1)(£-1)
χ Π 6-£(6-ΐ)£(6-ι) Π &
8=j + l
8=j + l
(6.1.6)
inserting (6.1.6) into (6.1.5) yields (6.1.4).

144
Chapter 6 Weak Convergence for Mixing Random Fields
Proof of Theorem 6.1.1.
Let ρ — p(n), q = q(n) be the positive integers such that
p, q —► oo, ρ — o(n), g = o(p)as η —► oo.
Denote
fc = fc(n) = [2n/(p + 9)],
^i(i) — [~™ + ip + *<3S -n + (г + l)p + гд] г — 0,1, · · · Д - 1,
£__г d times
In = (J 4(j), /^ = /η Χ /η Χ · ' · Χ /η ·
Then /^ consists of fcd d-dimensional cubes with side p. For given n, we
denote these к — к d-dimensional cubes by Δ^· , j — 1, · · ·, /с, i.e.
4d = U Δ5η)·
i=i
Put An — Jn\^n* By Bernstein's blocking technique, we need only to prove
1) a-2VarSAn-0,
2) Mk —► 0, X)j=i VarS(n, j) —► 1 and for any given ε > 0
Lk(e, <5) —► 0, as η —> oo.
Prom £|Xt|2+* < °°> E£Li^_V/(2+*)(r) < oo and Lemma 6.1.1, it
follows that σ2 is finite and VarS/ ~ σ2\Ι\ for any d-dimensional cube /
with |/| < oo. Therefore we have a~2Var5^n —> 0 and
к
]Г VarS(n,j) = Α:ση 2 Уаг5д(п)
/σ2(1 + ο(1)) -> 1, n-^oo.
J=i
^ JiJlt
(2n)dG2{l+o{l)Y
It also permit us to prove 2) only for the bounded random field (cf. Ibrag-
imov and Linnik 1971, the proofs of Theorems 18.5.3 and 18.5.4).

6.1 The CLT for mixing random fields
145
Now we prove M& —> 0. From (6.1.5) it follows that
k-i к к
щ < \^\EeitS(nJ) TT ett5(n,m) _ £jeitS(nJ)β ΤΤ eitS(n,m)
j = l m=j+l 771=7+ 1
fc-1
<^|^(eii5(n^-l-it5(n,j)
771=7+ 1
-£7(e<t5(nJ)-l-it5(n,i) + -5(n,i)2)£7 f[ eitS(n'TO>
m=j+l
fc —1 fc
+ |i|^|£5(n,j) ή eit5("'m)
j = l m=j+l
к
-ES(n,j)E [J eii5(n'm)
771=7 +1
+f £W(n,j)2 π ^5(n'm)
i=i
771=7 + 1
-ES(n,j)2E Π e*S(n,m)
m=j'+l
t2
=: Γι + |i|T2 + -T3.
(6.1.7)
Take ρ = o(n1/2), we have
l/l3 fc_1
Tr<2^Y:E\S(n,j)\3
3!
i=i
< C|t| (-) (Π ) ' ρ Ь5Д(П
η —> οο.
(6.1.8)

146
Chapter 6 Weak Convergence for Mixing Random Fields
Using the proof of inequality (6.1.4) we have
fc-i к к
T2 <σ~ιΣ Σ \ΕΧ* Π eits^^ - EXSE Ц eitS^^
ί=1 SGA(ti) m=7 + l m=j+l
3
k—1 к к
<ση-^ Σ Σ |^^s(eitS(n'm) - 1) Π eits(n^
j=l 8^д(п) rn=j+l r=m+l
к
- EXsE(eitS(n^ - 1) Ц eits(n^\. (6.1.9)
Г=771+1
Noting |XS| < Co, |eits("'m) - 1| < cpd/on a.s. and applying Lemma
6.1.1 in (6.1.9) we obtain
r2<cAn2E Σ Σ ν(^η'ΔΙη)))
<^/(1)Λ-2Σ Σ a(d(Af,A^))
j=l m=j+l
к
< cp2dnTdp-d Σ a(W, Δ}η)))
3 = 1
oo
/=1ί:(ί-1)ϊ<<ί{Δίη,,Δ<η))<ί,
OO
<cpdnrdY^ld-la{lq).
1=1
Prom condition (ii) of Theorem 6.1.1 it follows that
1=1
τ2<οΑτν(2τ+1)*Σ^ (6-1-Ю)
where β(η) — a(n)n^2r+1^d. Since /3(n) —> 0 as η —> 0 we can take ρ =
p(n) = o(n1/2), p(n) —> oo as η —> oo and g = g(n) —► oo, g = o(p) such
that the right hand side of (6.1.10) approaches zero.

6.1 The CLT for mixing random fields
147
Now we estimate T3.
Тз<а-2^ £ \EXSXU Π JtS(n'm)
i=1 s,ueA<n) rn=j+i
к
-EXaXuE J] eits(-n^
m=j+l
k—1 к к
^<2Σ Σ Σ \ExsXu(eits{n'j+1) -1) π eiiS(n,m)
j = l m=j + l sueA(n) m=j+2
- £XsXu#(e*i5(n'i+1) - 1) [J eii5(n'm)
m=j+2
„2d _d к-1 к
<^^Σ Σ v(^,a
j=l m=j + l
p2dnTd3(q)
* Cn<i/2q(2r;Z "> 0. as n->oo. (6.1.11)
Combining (6.1.7)-(6.1.11) yields that Mfc —► 0 as η —► oo.
Finally, we prove Ζ/&(ε,<5) —> 0. Noting that the stationarity of {£t}?
we have
Lfc(M) < cRVW) / |5д(п)|2^Р(сЬ)
Δι
<c(n^p)~d [ \SAin)\^P(cL·).
Ai
There exist ρ = p(n) —> 00, p(n) — o(n), and g = g(n) —> 00, g —
o(p) as η —> oo such that the right hand of the above inequality
approaches zero. This completes the proof of Theorem 6.1.1.
Remark 6.1.1. In the above definition of a*-mixing, the positions
of set U and V are symmetric, so the assumption:
OLm,n{r) < f(m)nTa(r)
in the condition (i) is not very reasonable. The random field {Xt?t € %d}

148
Chapter 6 Weak Convergence for Mixing Random Fields
is said to be α-mixing, if
a(r) = sup{|P(AB) - P(A)P(B)\ :Aeav,Be σν,
U,VcZd,d(U,V)>r}
-► 0 as r -► oo. (6.1.12)
A more general result was given by Lu (1995), which is a generalization of
Theorem 3.2.3 to α-mixing random fields.
Theorem 6.1.2. Let {Xt?t € %d} be an α-mixing random field with
EXt — 0, EX\ < oo. If there exists a g £ Q such that
oo
supEg(\Xt\) < oo, Σ^"1/*^)) < oo, (6.1.13)
lim Var5Jn/nd - σ2 > 0. (6.1.14)
Then
wn=>w
where Wn(t) = Sj[nt]/an, 0 < t < 1.
The proof of Theorem 6.1.2 is similar to that of Theorem 3.2.3.
Remark 6.1.2. A similar theorem for non-stationary random fields
has been discussed by Guyon (1992). The central limit theorem for a-
mixing random fields with continuous parameters has been discussed by
Gorodezkii (1984), Zhurbenko (1984), etc, and the weak invariance
principle for this case has also been given there.
Bradley (1992) proved the CLT of strictly stationary random fields
under the "unrestricted /o-mixing" condition and just finite or "barely
infinite" second moments. No mixing rate is assumed. Let {£t, t £ Zd} be a
strictly stationary random field. For any nonempty disjoint sets S,DC Zd,
put
p(S,D) = p(a(£k,k € S), a(Ck,k G £>)).
For each r > 1, define
p*(r) = sup p(S,D),
where the sup is taken over all pairs of nonempty disjoint subsets S,DcZd
such that d(5, D) > r. Denote
L:=L(") = (4"),...,/("))GNd, S(t,L)= £ ft.
l<t< L

6.2 Convergence of finite dimensional distributions
149
Bradley (1992) proved the following theorem.
Theorem 6.1.3. Let {Xt?t € %d} be a centered strictly stationary
random field with 0 < EXq < oo, p*(r) —> 0 as r —> oo and the continuous
positive spectral density /(·) on Td, satisfying /(1, ···,!) > 0, where Τ
denotes the unit circle in the complex plane. Then as \\LSn>\\ — l^ · · · r£ —>
oo one has that ||S(£, L)\\2 -> oo and S(£, L)/||S(f, L)\\2 -^ JV(0,1).
The proof of Theorem 6.1.3 will not be presented here.
6.2 Convergence of finite dimensional distributions
Denote
Jn = {} = tii/n>h/n>'~,jd/n) : JuJ2,~',Jd € {1,2,···,η}},
Cnj = (j-n-^J),
(a,b] = {{x\,X2,··· ,Xd) : a{ < X{ <bi,i = 1,2, — ,rf},
Г* = {(а,Ь]: a,bG[0,lf}.
Let {^njj JGj'n, n>l}bea triangular array field. It is easy to see
that a random field {£t5 t G Zd} on a d-dimensional integer lattice is
a special case of a triangular array random field {£nj, j G ,Τ^,η > 1},
if we write £nj — £nj, nj — (ji, J2, · · · ,jd). Now from a random field
{^njj j^^j^^llwe form the set-indexed partial-sum process of the
n-th level as
Zn(A) = n-d'2 £ !d^jl(inJ _ ££nJ) Ле^пёК, (6.2.1)
jeJ'n
l^nj,
where | · | is Lebesgue measure and Bd is the class of Borel sets of [0, l]d.
For the random field {^t,tGZ}, correspondingly, define
Zn{A) = n~d/2 Σ\ΑηCtl(£t - ВД (6.2.1')
tezd
where A/nd G Bd, Ct = (t - l,t),t - (ti,t2, ···,«*),** G Z.
In Sections 6.2 and 6.3, we prove the weak convergence of Zn to a
Wiener process W, restricting its domain of definition to a subset of Bd

150
Chapter 6 Weak Convergence for Mixing Random Fields
satisfying a metric-entropy bound. We also impose moment and mixing
conditions on the {£nj}. Let χ > 0.
Definition 6.2.1. The random field {£nj, j G Jn} is said to be
α-mixing if a{nx) —> 0 as η —> oo, where
a(nx) = sup sup \P(AB) - P(A)P(B)\.
IJCJn Aea(£ .j€/)
Definition 6.2.2. The random field {£nj, j G ^7n,} is said to be
p-mixing if p(nx) —> 0 as η —> oo, where
|Соу(Х,У)|
p[nx) — sup sup
/,JcJn XGL2(a(enJjg/)) \/VarXVary
d(/,J)>xyGL2(<j(CjjGj))
and i/2(^7) is the set of L2 random variables measurable with respect to
T.
Definition 6.2.3. The random field {£nj, j G Jn} is said to be
symmetric φ-mixing if φ{ηχ) —> 0 as η —> oo, where
<^(ηζ) - sup sup тах(|Р(Л|Я) - Р(Л)|, |Р(В|Л) - P(S)|).
/,JCJn AGa(^nJJG/)
P(A)P(B)>0
Definition 6.2.4. The random field {£nJ-, j G Jn} is said to be
absolutely regular if β{ηχ) —> 0 as η —> oo, where
/3(n*) = sup ||£(£nJ,JG/UJ)
/,Jcjn,rf(/,J)>x
-£(enJ,JG/)£(^j,JGJ)||var
£(£(·)) is the distribution law of {£(·)} and || · ||уаг ls variation norm.
It is clear that
a{nx) < p{nx) < 2φ(ηχ), α(ηχ) < β(ηχ) < φ{ηχ). (.6.2.2)
Next, we introduce the metric entropy condition. We say that Borel
sets Л, В in Bd are equivalent if | AAB\ — 0, and denote the set of
equivalence classes by £. Define di{A, В) — |ЛЛР|, it can be proved that c?l(·, ·)
is a metric on £. The set £ forms a complete metric space under d^.

6.2 Convergence of finite dimensional distributions
151
Definition 6.2.5. A subset A of £ is called totally bounded with
inclusion, if for every δ > 0 there is a finite set As С £, such that for every
A G A there exist Л+, A~ G Л5 with Л~ С А С Л+ and \A+ \ Α~\<δ.
Note that Д$ is a 5-net with respect to di for A
Let Л be a totally bounded subset of £. Its closure A is complete
and totally bounded, hence compact. Let C{A) be the space of continous
functions on A with the sup norm || · ||. Because A is compact, С (A) is
separable. Thus C(A) is a complete, separable metric space. Let CA(A) be
the set of everywhere additive elements of С (A), namely, elements / such
that f(AuB) = f(A) + f(B)-f(AnB) whenever Л,Б, AUB, ΑΠ Β G A.
It can be shown that for fixed ω,Ζη{·) G CA(A), i.e. Zn are random
elements of CA(A). A standard Wiener process on A is a random element
W of CA(A) whose finite dimensional laws are Gaussian with EW(A) —
0,EW(A)W(B) — \Α Π Β\. In order that W should exist it is necessary
(see Dudley 1973) that A satisfies a metric entropy condition.
Definition 6.2.6. Let A be a totally bounded subset of £, As be the
smallest <5-net of A. Denote
Ν(δ, A) - Card Λ, Η (δ) - log Ν(δ, Α).
A is said to satisfy a metric entropy condition, or a convergent entropy
integral, if
^(Щ))1/2^<оо. (6.2.3)
Define the exponent of metric entropy of A, denoted by r := inf{s, 5 >
0, Η (δ) = 0(δ~3) as δ -> 0}. If г < 1, then (6.2.3) holds.
Remark 6.2.1. Some examples of classes of sets which satisfy the
metric entropy condition are as follows:
If Cd denotes the convex subsets of [0, l]d, then r = (d — l)/2 (see
Dudley 1974).
If Id = {(a, b] : a, b G [0, l]d} as above, then r = 0.
If Pd'm denotes the family of all polygonal regions of [0, l]d with no
more than m vertices, then r = 0 (see Erickson 1981).
If £d denotes the sets of all ellipsoidal regions in [0, l]d, then r = 0 (see
Gaeussler 1983).
For the Vapnik-Cervonenkis class V that includes the above three
examples, it is known that N(6,V,d\) = Card V<$ < οδ~ν for some с and
υ > 0 (Dudley 1978).
When {&, t G Zd} is independent, the weak convergence of Zn to
W has been studied by Bass and Руке (1984, 1985), Alexander and Руке

152
Chapter 6 Weak Convergence for Mixing Random Fields
(1986), Lu (1992), etc. For the mixing random field {£П) j, j. G Jn, η >
1}, the weak convergence of Zn to W was first discussed by Goldie and
Greenwood (1986a, b). They proved the following theorem.
Theorem 6.2.1. Assume that Εξη^ j = 0, and
(i) for some s > 2, {|rad/2fnj|s, j G Jn, η > 1} is uniformly integrable;
(ii) the exponent r of metric entropy (with inclusion) of Λ satisfies
r < 1;
(Hi) β(ηχ) = 0((nx)b) as nx —> oo, the exponent b of absolute
regularity satisfies b > ds/(s — 2) and b > d(l + r)/(l — r);
(iv) the symmetric φ-mixing coefficients satisfy
oo
supyV/^'n-1) <oo;
n>lj=1
(v) for any null family {Dh,0 < h < ho} in Xd (a null family is α
collection such that Dh С D^ for h < h' and \Dh\ — h for each h),
lim lim sup
h[0 n—>oo
EZl{Dh) _ γ
\Dh\
0.
Then Zn converges weakly in CA(A) to W.
For a random field {£t>t € Zd}, the versions of Definitions 6.2.1, 6.2.2,
6.2.3 and 6.2.4 are as follows:
a(x)= sup sup \P{AB) - P(A)P(B)\,
/,JCZd A6a(ij,ie/)
d(I,J)>xB€a(^J€J)
( ч |Соу(Х,У)|
p{x) = sup sup
/,Jczd Χ<ΕΖ,2(σ(^,ί<Ε/)) VVarXVary
d(/ϊJ)>xУ€L2(σ(ξjj€J))
β(χ) = sup ||£(£j, j G / U J) - £($,, j G /)£«,, j G J)||Var,
/,JCZd, d(I,J)>x
φ{χ) = sup sup
d(i,j)>x Bea(^eJ)P(A)P(B)>o
χ тах(|Р(Л|В) - Р(Л)|, |Р(В|Л) - P(B)|).
Corollary 6.2.1. Let {£t?t £ ^d} be a strictly stationary real random
field with Εζ(0) = 0. Assume that

6.2 Convergence of finite dimensional distributions
153
(i) Ε\ξο\s < oo for some s > 2;
(ii) A has exponent of metric entropy (with inclusion) r < 1;
(Hi) β(χ) — 0(x~b) (x —> oo) for some b > max(ds/(s — 2),d(l +
r)/(l - r));
(iv) ЕГ=гР1/2(2^')<оо;
(ν) Σί6ζ-^οίί = 1·
Then Zn converges weakly in CA(A) to W.
Dobrushin (1968) showed that the (^-mixing condition is not satisfied
even for some simple examples of Gibbs random fields. Dobrushin and
Nahapetian (1974) introduced the nonuniform (^-mixing condition.
Definition 6.2.7. The random field {£t, t G Zd} is said to be
nonuniform φ-mixing, if for Лг С Zd, |Λί| < оо,г = 1,2, there exists a nonnegative
function </?|λ-||(·) depending only on |Λχ|, such that
sup \P(E\F)-P(E)\<<plAll(d(AuA2))
^σ(Λι),^σ(Λ2),Ρ(^)>0
and φ\Αι\(χ) —► 0 as χ —► oo, where |Λ| is the cardinality of Л.
Chen (1991) gave a sufficient condition under which a sequence of
partial-sum set-indexed processes with the nonuniform (^-mixing
condition converges to a Brownian motion when the indexed set A = Id —
{(a, b], a, b€[0,l]d}.
Theorem 6.2.2. Let {£t, t G Zd} be a strictly stationary nonuniform
φ-тгхгпд random field and satisfy
(i) there exists a non-negative function φ{·) on R1, such that for any
А С Zd, \A\ < oo, φ\\\(·) < \Α\φ(-), and for some δ > 0
limsup^r))1/2^4^ < oo, (6.2.4)
r—>oo
(ii) Εξ0 = 0, Ε\ξ0\2+δ < oo,
(Hi)
0 < °2 == Σ Cov(^05 £t) < oo. (6.2.5)
tGZd
Then Ζη/σ converges weakly in CA(A) to a Brownian motion as η —> oo.
By a direct calculation, Lu (1995) proved that the conclusion of
Theorem 6.2.2 holds for a more general indexed set A (with the metric entropy
exponent r, 0 < r < 1) and the condition for the rate of nonuniform φ-
mixing was weakened.

154
Chapter 6 Weak Convergence for Mixing Random Fields
Theorem 6.2.3. Let {£t>t € Zd} be a strictly stationary nonuniform
φ-mixing random field, and satisfy
(i) ψ\κ\{·) < \Μ<Ρ(·) o-nd 4>(x) = 0{x~2d-1-2dl6) for some 6>0,
(ii) ££0 = 0,£|£ο|2+δ<οο,
(Hi) A has a exponent of metric entropy (with inclusion) r < 1. Then
Ζη/σ converges weakly in CA(A) to a Brownian motion, as η —> oo, where
σ2 is defined as in (6.2.5).
Remark 6.2.1. From Theorem 6.2.2 a uniform central limit theorem
for certain Gibbs fields has been given in Chen (1991) which is also true
for indexed set A, if A has the exponent of metric entropy r < 1.
The proofs of Theorems 6.2.1 and 6.2.2 are omitted here.
The proof of Theorem 6.2.3 will need the following lemmas. Some
lemmas for moment estimations are of independent interest.
A slice in Rd is a set
5(c, α, η) = {χ G Rd : a < cxx < a + η},
where χ G i?d, |c| — 1, a G i?, and η > 0. The thickness is 77, the direction
(of the normal to the two bounding hyperplanes) c, and the displacement
a. The slice splits a set Л С Rd into three parts, namely Α Π 5(c, α, η) and
two sets
Л+=ЛП{хЕ^:с'х>а + η}, Α- = Α Π {χ G Rd : cxx < a}.
If A is measurable and |Л+| = |Л_| we say the slice bisects A.
Lemma 6.2.1. There exist Co^q, depending only on d, such that for
all ρ satisfying 0 < ρ < \/d, and for every measurable А С Rd of finite
measure, we can find a slice S that bisects A, has thickness (\A\/2)P, and
is such that
\A Π S\ < С0(|Л|/2)^+^/^+1). (6.2.6)
The proof of Lemma 6.2.1 was given in Goldie and Greenwood (1986b).
Let {Xh i G Zd} be a /9-mixing random field with EX\ = 0. Denote
00
i=0 >
Z(A)= Σ HnCi|Xi, A€Bd.
iezd

6.2 Convergence of finite dimensional distributions
155
Lemma 6.2.2. There exist constants а,Ъ, depending only on d, such
that
\\Z{A)\\2 < аеЧт^1/2, A G Bd. (6.2.7)
Proof. Without loss of generality we assume σ — 1. Set
a(h) = sup ||2(Л)||2, σ{Κ) = sup σ(Λ').
AeBd,\A\=h h'<h
Observe
\\z{A)\\2 < Σ HnCjHiXjib < £ μησ,ι = |л|,
so
σ(Λ) < h. (6.2.8)
Take ρ in Lemma 6.2.1 to be such that the exponent r — (q+pd)/(q+\)
does not have any positive integer power equal to 1/2. This is to avoid
a minor technically later. Pick A of positive finite measure 2m, say. The
slice S in Lemma 6.2.1 is S(c,a,rap) for some c, a. We refer to {x G Rd :
c'x > a + rap} and {x G i?d : c'x < a} as the "sides" of S containing A+
and Л_, respectively. Since |Л+| — |Л_| we know
m>\A+\ = \A-\>m-\AnS\>m- C0mr.
We may find Л+, such that Л+ is in the side of S containing Л+, is
disjoint from Л_, and has measure |Л+| — πι — |Л+|. Thus |Л+| < C§mT.
Let Л" — Л+ U Л+; then |Л+| = га. Similarly we construct Л'_ and A'L_
on the side of S containing A-. Now if χ G Л", у G Л" then
||χ - y|| > <Γ1/2|χ - y| > d-1'2™?.
Hence if C\ and Cj intersect Al\_ and Л", respectively, then
||i-j|| ><T1/2mp-2,
whence
£(Ζ«) + Ζ(^))2
< (1 + p(d-1/2mp - 2))(£Z2(A'|) + EZ2{A"_))
< (1 + p(d-1/2mp - 2))2σ(τη)2.
Since
Z(A) = Z{A%) + Z(A'L) - Z(A'+) - Z(A'_) + Z(A П 5) (6.2.9)

156 Chapter 6 Weak Convergence for Mixing Random Fields
the triangle inequality gives
σ(2τη) < 2χ/2(1 + p(d'^2mp - 2))1/2σ(τη) + Sa(C0mr).
Choose h > 1, then h = 2km where к G N and 1/2 < m < 1, and we have
σ(πι) < 1, a(2j+1m) < aja(2jm) + /3,·,
where
a,- := 2х/2(1 + p(d'1f22ipmp - 2))1/2, /3,- := Sa(C02jr).
Iterating,
fc-l fc-l fc-l
^)<IIai + Eft Π <*· (6·2·10)
j=0 j=0 i=j+l
Let / G N satisfy ρ > l/l. Now dll22jp2~p - 2 > 7? Iх for j > j0 > 1, so
oo oo
]T p{dTxl27?pmp - 2) < ]^р(2^)
3=3 о j=l
oo Ζ
j=0t=l
oo Ζ
Thus
oo
Π(! + p(d'^22ipmp - 2))1/2 < 2j°/2elp/2 =: αχβ6^, (6.2.11)
i=o
and (6.2.10) yields
fc-l
σ(Λ) < axeb^(2fc/2 + 3 ^ 2{fz'1'i^2'&{CQ2iT)). (6.2.12)
i=o
Let ν be the integer such that rv < 1/2 < r"-1 (recall that ru = 1/2
was excluded by choice of r). We shall use (6.2.12) iteratively, ν times, to
obtain the result of the lemma,
σ(Κ) < aebph1'2. (6.2.13)

6.2 Convergence of finite dimensional distributions
157
Applying (6.2.8) to the right-hand side of (6.2.12), if ν = 1 we find
a(h) < aieb^2k/2{l + 3C0(1 - 2'^2'г^)'г},
and since 2fc/2 < 21/2h1/2 the proof is concluded. In the other case, i.e.
when r > 1/2, we obtain
a(h) < aieb^2kl2{l + -Со(27"1/2 - l)"^},
whence a(h) < dx^xphT for some a[. Substituting this in (6.2.12), if
r2 < 1/2 we obtain (6.2.13) and otherwise a(h) < a2eb2phr2. After a total
of ν uses of (6.2.12) we obtain (6.2.13).
By the same way, we can prove the following lemma.
Lemma 6.2.3. Suppose that τ :— supj \\Xi\\3 < oo for s = 2 + <5,0 <
δ < 1 and
oo
ρ" = Σ/*2/ί(2ί)<°°·
г=1
Then for any A £ Bd we have
\\Z{A)\\s<cecP\\A\ll2. (6.2.14)
Proof. First it follows from Lemma 6.2.2 that Lemma 6.2.3 holds
true for δ — 0. For the case of 0 < δ < 1, let
r(m) = sup ||Z(^)||S, т(т) — sup т(га').
Ae#d,|A|=m m'<m
From (6.2.9), we obtain
r(2m) < ||Ζ(^|) + Ζ(^)||2+« + 3r(C0mr). (6.2.15)
By using
|1 + χ\2+δ < 1 + 9|x| + 9|z|1+* + \χ\2+δ, (6.2.16)
we have
E\Z{A'[) + Z{A'L)\2+8
< 2τ2+δ(πι) + ${E\Z{A'{_)\\Z{A'L)\l+8
+ E\Z{A'[)\l+6\Z{A'L)\). (6.2.17)

158 Chapter 6 Weak Convergence for Mixing Random Fields
By Lemma 1.2.7, we have
E\Z(A'l)\\Z(A'L)\1+s
< E\Z(A'[)\E\Z(A'L)\1+S + 4рШ(а~12тр - 2)r2+6{m)
< (l + 4p^b(d~5TOP _ 2))τ2+δ(τη). (6.2.18)
Similarly we have
E\Z{Al)\x+e\Z{A'L)\
< (l + 4p*h(drbmP - 2))τ2+δ(ιλ). (6.2.19)
Inserting (6.2.18), (6.2.19) into (6.2.17) yields
\\Z(A'l) + Z(A'L)h+s < 202b (l + 2p£*(dr?m* - 2))^r(m).
Write po = p2l{-2+s){drll2mP - 2), whence
r(2m) < 202^ (l + 2ро)Шт(т) + Зт(С0тТ).
Iterating, for h = 2km, к € Ν, 1/2 < m < 1 we have
fc-l fc-l fc-l
r(h)<l[aj + Y/0j J] <*ύ
j=0 j=0 i=j+l
where
aj < 202b{l + 2p£s(d-hjpmp - 2)}Ш, β5 = 3r(C02jrmr).
The conditon p" < 00 implies that
00
Σ Ρ^~δ (d-hjpmp - 2) < 00,
3=3 о
where jo is defined as in Lemma 6.2.2. The remainder of the proof is the
same as in the case of δ — 0. The proof of Lemma 6.2.3 is completed.
Denote
oo
Ρ' = Σ,Ρ1/2(ή
;=o
g(y)=supEX?I(\Xi\>y).

6.2 Convergence of finite dimensional distributions
159
Lemma 6.2.4. Let {Xj, i G Zd} be a p-mixing random field. If
{Xj2, i G Zd} is uniformly integrable and p' < oo, then the set of
random variables {Z2(A)/\A\,A G Bd, \A\ < oo} is uniformly integrable, i.e.
there exists a constant С depending only on d, such that
Е^Ж1^Щ1>У)) < (C(1 Α^) + ^(^)}β^. (6.2.20)
Lemma 6.2.5. Let {W(A), A G Bd} be an additive processs that
satisfying
(i) EW(C) = 0 for any С G Id,
(ii) EW2(C) = \C\ for any С G Id,
(Hi) W(Ci), · · ·, W{Ck) are independent whenever Ci, · · ·, Q. G Id
and d(C{, Cj) > 0 for г ф j,
(iv) Urn™-*, EjeJm Pi\W(Cmj)\ > *} = 0 for any ε > 0.
Then W is a standard Wiener process on 1Z = 1Z(Td)(the ring of all finite
unions of elements ofXd).
Lemma 6.2.6. Let {Zn(A), AeTl = Tl(Id), η > 1} be a sequence of
additive processes such that
(i) EZn(C) -+ 0 (n -> oo) for any С G ld,
(ii) EZ2n{C) -> \C\ (n -> oo) for any С G Id,
(Hi) whenever Ci, · · ·, Ck Gld are such that p{C{, Cj) > 0 for г Ф j
we have for all real z\, · · ·, Zk that
к
P{rft=1{Zn{Ci) < Zi)} - Ц P{Zn(Ci) < z{} -^ 0, as η -+ oo, (6.2.21)
t=l
(iv) limm^oo limsup^^ EjeJm p{lZn(Cmj)l > ε} = 0 for any ε >
0,
(υ) for each С G Id, the set {Z%(C),n > 1} is uniformly integrable.
Then the finite dimensional distributions of Zn converge weakly to the
corresponding finite dimensional distributions of a Wiener process W on 1Z.
Lemma 6.2.7. Let {Zn} be a sequence of additive processes on Bd such
that the set {Z2(A)/\A\, A G Bd,n > 1} is uniformly integrable, and
satisfying Lemma 6.2.6 (i), (ii), (Hi). Then the finite dimensional distributions
of Zn converge weakly to the corresponding finite dimensional distributions
of a Wiener process W on Bd.

160
Chapter 6 Weak Convergence for Mixing Random Fields
The proofs of Lemmas 6.2.4—6.2.7 are given in Goldie and Greenwood
(1986 a, b).
Theorem 6.2.4. Let {£nj j G Jn}, be a p-mixing triangular array.
The smoothed partial-sum processes Zn are defined by (6.2.1). Suppose
that
(i) ££nJ = 0 for any n>l,jG Jn;
(ii) the set {ηάζ^ :,n > l,j G Jn} is uniformly integrable;
(Hi) supn Σ(^=1 pn (n~12·7) < oo, and Lemma 6.2.6 (ii).
Then the finite dimensional distributions of Zn on Bd converges weakly to
the corresponding finite dimensional distributions of a Wiener process.
Proof. By Lemma 6.2.4 we have uniform integrability of the set
{Z2(A)/\A\, η > 1, A G Bd}. Because pn(x) is non-increasing in ж,
condition (iii) implies pn(x) —► 0(n —* oo) for each fixed x. Thus an(x) —► 0(n —►
oo). Clearly the left hand side of (6.2.21) does not exceed (k — l)an(#),
where θ > 0 is the least separation distance between the sets Ci, · · ·, C^.
Hence Lemma 6.2.6 (iii) is satisfied. By Lemma 6.2.7, the proof of Theorem
6.2.4 is completed.
6.3 Tightness
Fisrt we present a lemma for nonuniform (^-mixing sequence, which is
similar to Lemma 6.2.3.
Lemma 6.3.1. Let {&, t G Zd},{Zn(A),A G Bd} be as in
Theorem 6.2.3 with φ(χ) = 0{χ-(ά+χ+θϊ), for some θ > 0, instead of φ{χ) —
0(x-(2d+l+2d/6)y Then for any AeBd
\\Ζη(Α)\\2+δ < caolnA]1/2 (6.3.1)
where σ$ = Εξ^.
Proof. First, we prove that for δ = 0
σ(2τη) < 2χ/2(1 + 2πι1Ι2φ1Ι2{ά-1Ι2τηΡ - 2))1/2σ(πι) + Щстг),
where σ(τη) = supi4€fid||i4|=m ||2П(Л)||2, σ(τη) = s\iprnf^rna(mf). Take
0 < ρ < 1/d in the bisection lemma to be such that the exponent r =
(q + pd)/(q + 1) does not have any positive power equal to 1/2. For any
given h > 1, we write h = 2fcm, к G N, 1/2 < m < 1. Note that σ(πι) < 1,

6.3 Tightness
161
and for nonuniform (^-mixing, by the same discussion as in the proof of
Lemma 6.2.3 we have
a(2j+1m) < aja(2jm) + /3,·,
where
aj = 2χ/2(1 + 2 · y/2m*/Md-1/22jprnp - 2))1/2,
βΐ = 3 a(c2js).
Iterating,
k—X k~l k~l
σ(Λ)<Π«; + Σ# Π <*■ (6·3·2)
j=0 j=0 i=j+l
1 1 q+pd
Furthermore, we take p, - > ρ > -—-, such that r = — does not
a d+1 q+1
also have any positive power equal to 1/2. There exists a jo > 1 such that
di/22jp2-P > 2i/(d+i) for j > j0, so
oo
ip := £ VfitpWidT1'2**™? - 2)
3=30
oo
<^2^V1/2(2j/(d+1))
<с^222-2-5Фт(<г+1+е)
OO _ ·
= c ^(2^/(2(^+1))) '<QO
3=1
Thus we obtain
fc-i
σ(Λ) < cec^(2t +3^2-Ha(c2jr)). (6.3.3)
j=o
The remainder of the proof is the same as in the proof of Lemma 6.2.2.
This proves that (6.3.1) holds for 6 = 0.
Consider the case of 0 < δ < 1. Recall the notation of Lemma 6.2.2
and write
Zn(A) = Zn(A'[) + Zn{A"_) - Zn{A\) - Zn{A'_) + Zn(A η 5), (6.3.4)
where |Л| = 2m,S is a slice of А, |Л"| = \A'L\ = m,A+ and A'L is
situated on the different side of 5, the separation distance d(A'+,A'!_) >

162
Chapter 6 Weak Convergence for Mixing Random Fields
d ll2mp, \A'+\, \A'_\, and А П S all do not exceed cmr. Denote r{h) —
supiAHViee·1 Ι|Ζη(Λ)||2+«, r(h) = suph,<hr(h'). Prom (6.3.4),
r(2m) < \\Zn(Al) + Ζη(Α"_)\\2+δ + 3r(cmr). (6.3.5)
By (6.2.16) we have
E\Zn(Al) + Zn(A'L)\2+s
<2τ2+δ(πι) + 9(Ε\Ζη(Α'1)\\Ζη(Α'ί)\1+δ
+ E\Zn{A'[)\x+6\Zn{A"_)\). (6.3.6)
By the property of nonuniform (^-mixing, we have
E\Zn(Al)\\Zn(A'L)\1+s
< E\Zn(A'i)\E\Zn(A'L)\1+s + 2ψ^{ά-\τηΡ - 2)т2+6{т)
<(1 + 2т^Ш(Г2тр- 2))τ2+δ(πι). (6.3.7)
Similarly we have
E\Zn(Al)\1+6\Zn(A'!_)\
< (l + 2πι^φ^(d-2mp - 2))r2+<5(m). (6.3.8)
Inserting (6.3.7), (6.3.8) into (6.3.6) yields
\\Zn{A'[) + Zn{A'L)\\2+6 < c(l + m^(f2h(d-12mi> - 2)) Шт(т).
Whence
r(2m) < c(l + πι^φ^((1'^πιρ - 2)) 2+δ r(m) + 3cr(cmr).
Iterating, for h = 2fcra, к £ N, 1/2 < m < 1 we have
fc-i fc-i fc-i
τ(Λ)<Παί + Σ& Π <*>
i=0 j=0 i=j+l
where
Note
oo
>Г 25+Ур5ТУ (сГ *2"W - 2)
OO
c£(2«/(2(2+*)(«H-i)))-'<00#
J=JO
00
<
i=i

6.3 Tightness
163
The remainder of the proof is the same as in the case of δ = 0. The proof
of Lemma 6.3.1 is completed.
Proof of tightness in Theorem 6.2.3. For / G C(A) write the
modulus of continuity
"«(/)= sup \f(A)-f(B)\. (6.3.9)
AyBeAy\AAB\<6
Then, since A is compact, we can use a version of the Arzela-Ascoli
theorem: a subset U of С (A) has a compact closure iff it is equibounded
(supfeUsupAej\f(A)\ < oo) and equicontinuous (limsiosupfeUu6(f) =
0). Using this, from Theorem 8.2 of Billingsley (1968) it follows that a
sequence {Zn} of random elements of C(A) is relatively compact, i.e. every
subsequence of {Zn} contains a weakly convergent subsequence iff
(a) for each element A of some countable dence set in A, the family
{Zn(A),n > 1} is tight, and
(b) for every λ > 0, lim^jo lim suPn->oo Ρ{ω(Ζη) > λ} = 0.
(a) follows from Theorem 6.2.4. We need only prove (b). Without loss
of generality, we assume that σ2 = 1, Εξι = 0. For 0 < и < ν < oo, define
Vnj(u,v) = n^kniI(u < η^/^+^η-^Ι^Ι < t;) j G Jn,
Zn(A,u,v)= Σ \r V (nnjfav) - Εηηιί(ιι,υ)).
ieJn l n^
Lemma 6.3.2. Suppose {£t? t G Zd} satisfies the conditions of
Theorem 6.2.3. Then as η —► oo, Un(Id,a, oo) —► 0, a.s., EUn(Id, a, oo) —► 0,
where Id = [0, l]d, a > 0 and
Un(A,a,oo)= Σ \r ?J Ι^(α»οο)Ι-
Proof. We use Bass's technique (1985). For к = (fcb · · ·, kd) G Ъ%,
let /i(k) = max{fei,"-,fed} and Φ(ί,α) = sup{A: G Z+ : afcd/(2(1+<5)) <
г + 1}. It is well known that Card{k G Ъ% : /i(k) = r} < Crd~x, where С

164 Chapter 6 Weak Convergence for Mixing Random Fields
is a positive constant depending on d.
£ /(г + 1 > а/,(к)<*/(2(1+*)))М(к)-^2
kez^
oo
= Σ Σ l{i + l>ardl^^ydl2
oo
= c^/(i + l>ar*(1+{»)r(d-2)/2
<c(*(i,a))d/2<ca-1(i + l)1+*.
Thus
Σ £|£k|/(a(/,(k))W+*)) < |&| < oo)(Mk))-d/2
kez^
< Σ Σ (< + 1)^0 < Ifkl < i + 1}ШГФ
к t+l>o(M(k))d/(2(1+*))
oo
* Σ[Σ'(«+ ! > a(M(k))d/(2(1+5)))(Mk))-d/2]
<=o к
• (t + l)P{t < \ξο\ <i + l}
OO
< ca~x £(» + l)2+*p{i < |fo| < » + 1}
i=0
< co-^deol + 1)2+* < oo.
Then for any ε > 0 and almost sure ω, there exists an η\(ω), such that
Σ |£к/(«Мк)^1+й» < |&|< oo) |/(м(к))</2 < ε.
k:^(k)>ni
From this,
Ε/η(Λα,οο)< Σ |&|/(an<|/«1+*» < |&| < oo)n~d/2.
M(k)<ni
Thus, as η —* oo, Un(Id, a, oo) —* 0, a.s. Analogously, we can obtain
EUn(Id, a, oo) -+0.
Lemma 6.3.2 is proved.
In order to prove (b) we need only prove
limlimsupP{||Zn||^ > λ} = 0, (6.3.10)
i/jO n—>oo

6.3 Tightness
165
where Av = {A\B : A, B £ A, \A\B\ < u}. Since Zn(A) = Zn(A,0,a) +
Zn(A, a, oo), and
\Zn(A, a, oo)| < Un(Id, a, oo) + EUn(Id, a, oo),
by Lemma 6.3.2, in order to prove (6.3.10), we need only prove
limlimsupP{||Zn(^,0,a)||^ > λ} = 0. (6.3.11)
i/jO n—►oo
Let pn = [n(2+*)/(2(i+*))] and mn = n/{2pn). We divide Id in the
following two ways: CPn)b 1 G JPn and C2pn)i, 1 G J2pn. There are 2d C2pnl
in each Cpnl. Denote by /П)М the ith C2pnJ in CPnyU 1 G JPn,j G J2pn.
Let
Then
2d
Zn(·, 0, a) = ]Γ Ζη(· Π /п,г, 0, α).
г=1
Now in order to prove (6.3.11), we need only prove
limlimsupP{||Zn(^n/n)bO,a)||^ > λ} = 0. (6.3.12)
i/j0 n—►oo ^ J
Write
Ζη(ΑΠΐη^α)= Σ Σ ΙΛ П ^ ? Cnjl (^nj(Q^) ~ ^ηϋ(Ο,α))
lGJPnJG5(n,l,0 I "J1
= : Σ ^п1(ЛП/П|<,0,а),
leJPn
^(0,α) = η-ί^^/^/^+^η-^Ι^Ι < a), j G Jn,
where 5(п,1,г) = {j G Jn : /пД>< П CnJ ^ 0}. Denote Vnl = Vnl(A П
/П)г, 0, a). By the property of nonuniform (^-mixing, we have
£eaEk€JPn vnk <EeaV^Eea^k*\,keJPn Vnk
+ 2^5(nW|(^)^^
Note that \Vn\\ < 2a and eaVnl < 1 + аУп1 + a2V^ when аа < 1/4. From
Lemma 6.3.1 it follows that
£7^<С|ЛП/П>1><| for i=l,2,...,2ii.

166
Chapter 6 Weak Convergence for Mixing Random Fields
Therefore for aa < 1/4
Ee*vni < eEa2Vni < eCa^AnI^, (6.3.14)
Inserting (6.3.14) into (6.3.13) yields
Eea^\eJPn УгЛ < wEea^k*\, keJPn Vnk?
where
w < eCc?\AnlnXi\ Л + 2el/2|5(n> ^ qUJL))
V V 2pn//
V ^ (2pn)^V2pn^
Iterating this procedure, we have
Ее«ЕшРп у* < ес«Члтп,<\Л + 2ei/2^i_С^МГ«
= ехр(са2|Л| + сп~^^+^)
<с ехр(а2|Л|) (6.3.15)
for large п.
Now we return to estimate the left hand side of (6.3.12). Since 0 <
r < 1, we can take 5 > 0 such that r < 1/(1 + 5). Set
δ3 = ν/ν, j = 0,l,···,
Aj = Аое-*1+г-р<2+*»/<2+'> j = 1,2, · · ·,
A0-A(l-2-(1+'-r(2+s»/(2+i)),
aj = e-^+-)A2+')a, j = 1,2,.·.,
a = «/!/(!+.), Co = (6Ε|ξ0|2+7λο)1/(1+ί).
For any Л € A) there exist Aj, Aj € A)(^j) such that Aj C. AC. A+ and
|Α^\Α,·| <£,·. Then
Ζη(ΛΠ Jn>i,0,a)
OO
= Ζη(^οΠ/η)ί,0,α) + ^{Ζη(Α^+ιΠ/η)ί,0,α^) - Ζη(^· Π/^,Ο,α^·)}
i=o
00
) - Zn(Aj n/n)baj,aj-i)}.

6.3 Tightness
167
So if \\Ζη(· Π /П)г, 0, α)||л0 is to exceed λ, at least one of the following must
hold:
(a) for some A0 G Λο(δ0), \Zn(A0 П Jnji, 0, a)\ > λ0;
(b) for some j, for some
Aj G Ao(6j),Aj+1 G A(£j+i), Ι4?δ4?+ιΙ < 2«j,
\Zn(Aj+i П Jnji,0,aj) - Zn(i4j Π Jnji,0,aj)| > 2λ^;
(c) for some j, for some
л,·,л;gл№), ^асл;, |л;\л,-|<«,·,
|Ζη(^4 n/n)i,aj,aj-i) - Zn(4j Π /η^,α^,α^·_ι)| > Aj.
The number of pairs Aj^A^ in Aq{6j) is < ехр(4Я(^/2)), while the
number of pairs Л, G A>(£j), Л7+1 G .4o(£j+i) is < ехр(4Я(^+1/2)).
We have
00 00
P{||Zn(.n/n)b0,a)|U0>A}<p0 + Eri + E5^
j=o 3=1
where
Po <2exp{2H(60/2)} max Ρ{|Ζη(Λ> П Jn>i,0,a)| > λ0},
|Λ0|<2(5ο
г,- <4ехр{4Я(^+1/2)} max (P{\Zn((Aj+1\Aj) П /„,»,0,а)| > λ,·}
|Α,·+ιΔΑ,·|<2«,·
+ P{|Zn((Ai\Ai+i)n/n,i,0,a)| > λ,·}),
Sj < exp{4if(&,/2)} max P{ sup \Zn(A Π/η j,aj,aj_i)
А,СА+,|Л+\а,-|<26,· А,СДСЛ+
- Zn(Aj П /nii,aj,ay_i)| > Aj}.
By (6.3.15), taking a = l/4ao, we have
po < 2exP{2H(6o/2)}exp(-p- + <*%)
V 4a0 ag/
<2exP{C2^4-r-^ + ^}·
Similarly
rj<

168
Chapter 6 Weak Convergence for Mixing Random Fields
Thus
j=0 j=l
λ,
7=0 ^ ^
oo
i=o
Because of r < 1/(1 + 5), the coefficient of 2гг may be negative by choosing
ν small enough, and (6.3.12) follows. Theorem 6.2.3 is proved.

Chapter 7 The Berry-Esseen Inequality and
the Rate of Weak Convergence
It is well-known that the uniform estimation of the difference between
the distribution function Fn(x) of the normalized sum of the first η terms
of the sequence of independent random variables {Xn,n > 1} and the
normal distribution function Φ (χ) is given by the Esseen and the Berry-
Esseen inequalities. Furthermore, there exists a succinct non-uniformly
estimation
\Fn(x) - Φ(χ)| < Apo/(yfc(l + \x\3))
for the case of i.i.d. random variables {Xn} with i£|Xi|3 < oo, ΕΧχ — 0,
where p0 = Е\Хг\3/а3,а2 = EXf,Fn(x) = P{Sn/ajn < x).
In this chapter, we shall give the uniform estimations for α-mixing and
p-mixing sequences in Section 7.1
In Section 7.2 we shall discuss the Prohorov distance L{PoW~x^W)
between the measure Wn generated by the partial sums processes {Wn(£), 0 <
t < 1, η > 1} and the Wiener measure W. We shall give the estimation
of L(P о W~x, W) for a <£>-mixing sequence.
7.1 Rate of convergence in distribution
for α-mixing and p-mixing sequences
The proofs of the Esseen and the Berry-Esseen inequalities for the
independent random variables are based on the following proposition (Petrov
1975):
Proposition 7.1.1. Let Fn(x) be a distribution, fn{i) be the
characteristic function of Fn(x). Then for any given Τ > 0 and b > 1/(2π) we

170 Chapter 7 The Berry-Esseen Inequality and the Rate of Weak Convergence
have
rT , f (t\_ e-t2/2 л
вир|^п(*)-Ф(*)|<ь/ ψ±±- \dt + 1{h)^=- (7.1.1)
χ J-τ1 t Ι ν2πΤ
for some 7(b) > 0.
To obtain the estimation of |/n(0 — e~* ^2| is the key to the proof.
It is simpler in the case of independent random variables. How to get
the estimation of \fn(t) — e~l ^2| for the case of mixing dependent random
variables ? A method was given by Tikhomirov (1980), a modification of
which was given by Sunklodas (1984). The sketch of this method is as
follows:
Let {Xn,n > 1} be an α-mixing sequence with EXn — 0. Denote
σ2 = ES2n, Zn = 5η/ση, Fn(x) = P(Zn < x).
Assume that 1 < h <n — 1, 2 < к <n — 1 such that 2kh + 1 < n. Put
Л0)
3
rV) _
Zj — Xj/an — Yji
\p-j\<lh
(0)
Z3
β)- 7. _ zO
Λ») _ 7
Zi — ύηι
ζ}" = Zn - Z)l>, (7.1.2)
^^expM^-1)-^)}-!,
1=1
(r) -itzV л
Vj-e 3 ~ 1,
/n(i)=^eftZ»,
for j = 1,2, · · ·, n; / = 1,2, · · ·, r — 1; r = 2, · · ·, k. Since
/η(ί) = <.Σ EYjeitZ« = i Σ EY3eitzi
3=1 3=1
(o)
and
itz^
Eextzi = E(V)r> + 1)/η(ί) + Ε[(η)τ> - Εη)ν>)ε«ζ«]

7.1 Rate of convergence in distribution for a-mixing and p-mixing sequences 171
for j = 1, 2, · · ·, n; r = 2, · · ·, /с, the derivative of the characteristic function
fn(t) has the representation :
/„(o = ifiX1^50 + υ + ΣΣ4~1)Μ) + !)}/п(о
j=l r=3j=l
+ * Σ t{m Π ^еЧ}) - ВД П ^)Ее< }
r=2j=l ί=1 ί=1
+*ΣΣ4Γ_1)£{(^)-4ν^}
r=2j=l
+ г Σ EY3eitzf + i Σ EYj [J efeiteib). (7.1.3)
3=1 3=1 1=1
If we can give the estimation for each term of the right hand side of (7.1.3)
under some suitable conditions, then we get the differential equation
f'n(t) = (-t + a(t))fn(t) + b(t). (7.1.4)
By solving this differential equation, one obtains the estimation of \fn(t) —
e~l /2|. Finally, from (7.1.1) the estamation of Δη — supx \Fn(x) - Ф(ж)|
follows.
Denote 5 = 2 + δ. Assume that
d := max Ε\Χά\2+δ < oo, 0 < δ < 1, (7.1.5)
\2+δ
г
σΐ := £S^ > con for 0 < cq < oo. (7.1.6)
Sunklodas (1984) gave the following theorems.
Theorem 7.1.1. Let {ХП5 η > 1} be an α-mixing sequence with EXn =
0, and satisfy conditions (7.1.5), (7.1.6) and
a(n) < Ke~Xn λ > 0, К > 0.
Then there exist c\ = ^(Κ,δ), c<i = C2(K,6) such that for λ, λχ < λ <
λ2,
d (\og{an/cll2)Y+6
An-Cl^n( λ ) ' n~h (7-L7)
where
λχ = c2(log(an/cl/2))b/n, b > 2(1 + δ)/δ;
X2 = 4(2 + 6)6-1log(an/c10/2).

172 Chapter 7 The Berry-Esseen Inequality and the Rate of Weak Convergence
Theorem 7.1.2. Let {Xn,n > 1} be as in Theorem 7.1.1, but
a(n) < Κη~μ, μ = 2/3(1 + δ)(2 + δ)/δ2, β > 1, Κ > 0.
ΤΤιβη there exists а С — Ο(Κ,β,δ) such that for every η > 1
Δη < Cdc-^+^^/^^a-^-1)5/^^). (7.1.8)
Remark 7.1.1. Theorem 7.1.1 implies the Theorem 2 of Tikhomirov
(1980). Theorem 4 of Tikhomirov (1980) also points out that if £|Xi|4+7 <
oo, 7 > 0, then
Δη < en-1'2 log n. (7.1.9)
The proofs of Theorems will use the following lemmas.
Lemma 7.1.1. We have
ijr*™ = -* + ®20(1 + 12а)г'2<Р'*ап(а(Г1 + 1))(—2>/2β |*|/og/2
+ θ(23~7(5 - l))d(2h + 1Г^rV^r2), (7.1.10)
w/iere α = Ej=i(a0))(s_2)/s5 s = 2 + £, 0<£<1 and Θ is a constant
with [θ\ < 1.
Proof. Let r be a random variable uniformly distributed on the set
{1,2,···, n}, and independent of {Χι, Χ2, * * * 5 Xn}· It is easy to see that
for г = 0,1, · · ·, к and any b > 1
η
£|Ζ«|6<(2Μ + 1)6^£|^|6/η. (7.1.11)
i=i
By the Holder inequality and (7.1.11), we obtain
f>(|y,||zj V1) < 4I>4I J4x)lirx
71
< (2h + Ι)*"1 ΣE\Yj\S- (7.1.12)
Denote by ZjW the sum of those Yp from 2J· % for which ρ < j — Z/ι, and by
ij-O the sum of those Yp from Zj \ for which ρ > j + //ι. Thus 2J· ' = i}^ +

7.1 Rate of convergence in distribution for a-mixing and p-mixing sequences 173
Zj(l\ From Lemma 1.2.4 it follows that
\EYjzjW\ < Wiaih + ltf-b-illYjU^h
< 10(1 + 12aY>2d2ls{a{h + 1))(-2)/27(^/2σ„).
The same quantity also bounds \EYjZj^'\ for j = 1,2, •••,n. Since
Σ]=ι EYjZn = 1, we have
£ EY}zf = 1 - ±(ΕΥ3ζψ) + Щгр))
3=1 3=1
= 1 + Θ20(1 + 12a)1/2d2/s
x(a(h + l))(s-2V2san/c30/2, (7.1.13)
where we have used (7.1.6). According to Taylor's formula
*Σ a? = ~ Σ ΕΥίζΤι + θ(23~7(* -1)) ΣΕ{\γ5\\ζψγ~^)\ιγ-\
j=l j=l j=l
Inserting (7.1.12) and (7.1.13) into this equation and using (7.1.6), we
obtain (7.1.10).
Lemma 7.1.2. For \t\ < (an/32d1/s(h + 1)) =: Tu j = 1,2,···,η
and r — 3,4, · · ·, k, we have
\α{;~1}\ < ATd3ls{h + 1)2<2(1/2)47σ£
d>ls, ,. _,._,w.r /l\r
+
c—(a(h + l))(e-2)/s \r (-Y + r4r(a(/i + l))1/*
σ„ L V2/
=: α(Γ-χ) (7.1.14)
and
|41)|<2d2/s(/l + l)|<|/a2. (7.1.15)
Proof. Note that ajr_1) = e[y3 ЩГ* if}· From
\а<Я\ = \Щ(Р>\<ЩЕ\Ха\ Σ \χρ\/°1
\p-3\<h
<2d2l*{h + \)\t\/a2n,
(7.1.15) follows.

174 Chapter 7 The Berry-Esseen Inequality and the Rate of Weak Convergence
Now, to prove (7.1.14). Denote
j-(J-l)*
$ = exp{it Σ Υρ} ~ !>
p=j-lh+l
j+lh
$=exp{ii ^ Ур}-1.
ρ=ί+(ί-1)Λ+1
Obviously
It follows that
i(0
(Οι
£0)|
кл<ет+ет
r-1
ι-ΓΊ<ψ,·Π<
(0
ί=1
r-1
r-1
<ΣΣ*%Π^ Π &
(ίμ)
k=0
ι/=1
(7.1.16)
μ=&+1
where the summation Σ* sums up for 1 < l\ < · · · < /& < r — 1, 1 < lk+i
< · · · < /r_! < r — 1, \v φ\μ{ν φ μ). For each term of the right hand
side of (7.1.16) we have
4ШЙ0 Π Φγ
ι/=1
2+6 1+6
7Λ)πΛ)ί+«\2+ί
μ=&+1
ν μ
•Mirtfwn*
(7.1.17)
where Π' is the product for all even /, Π" is the product for all odd /. It
is easy to see that
4/2
ν μ
+ cr2r-l(a{h + 1))^ (Ε\Υά\2+δ)ϊΤ6,
*iir#ww
(7.1.18)
< crV^aQi + 1) + Π'^Ι^^+Ή'^Ι^^Ι2^ (7-1.19)

7.1 Rate of convergence in distribution for a-mixing and p-mixing sequences 175
and
max
:(e\$>\™, Εΐφ^*) < (№)2+δ л 22+*,
where bh = maxi<i<h+iГС*=1Ху||о±χ· ^ follows from (7.1.17)-(7.1.19)
that we have
2+δ
%-П^Ш^
•J'2
<2Td^s(h + l)2t2(l/2)4r/al
SI
-{a{h + l))->\r{
+ c—(a(h + 1))^ \r(l)2r + r2T(a(h + l))1/*!. (7.1.20)
Note that the number of terms of the right hand side of (7.1.16) does not
exceed 2r. (7.1.14) is proved.
We shall use the following results: for any finite ρ > 1
Y^rp/er < oo.
r=l
And for j = 1, · · ·, n; r — 2, · · ·, к
{Е\г,^\')У> < \t\dx's{2rh + 1)/ση.
Assume that 1 < /ι < η — 1, 2<fc<n—1 and
k3/4k(a{h + l))1/s < 1.
(7.1.21)
(7.1.22)
(7.1.23)
Lemma 7.1.3. If (7.1.23) is satisfied and \t\ < Tb Then
к
|Σ4υ^ + ΣΣ«Γυ^Γ + υ|
j=l r=3j=l
< - [d3/s(/i + l)2t2/an + dllscrn{a{h + l))(s"2)/s], (7.1.24)
CO
Σ|Σ4Γ_1)^-^ν<ζι
r=2 j=l
< JL[(fc + l)1/2 + cVWih + l)2t2/a2
co
(7.1.25)

176 Chapter 7 The Berry-Esseen Inequality and the Rate of Weak Convergence
In addition, if к > (logn)/81og2, then
Ё№ШГ*'"!Ч)1
j=l 1=1
<cc^'2dzls{h + l)2t2/a2n
(7.1.26)
Proof.
к п
|Σ41)^2) + ΣΣ4Γ"1)^1Γ) + 1)
j=l r=3j=l
3=1 r=3j=l
which implies (7.1.24) by using Lemma 7.1.2, (7.1.22), (7.1.23) and (7.1.6).
Further, Cov(£,77) = Ε(ξ — Εξ)(η — Erj). Applying the Holder
inequality and Lemma 1.2.4, we obtain that for r = 2,3, · · ·, к
Σ«Γχ)φίΓ) - 4VZn
3=1
1/2
<{(Σ Σ +Σ Σ )a(f-4-1)coy^\v^)}
j=1 \P-J\<2rh 3 = 1 \p-j\>2rh
<{Σ Σ l^lK-^IH^Ibll^JIb}
i=1 |p-il<2r/i
+{24Σ Σ ι«5'~1>ιι«ίΓ~1)ι
j=1 \p-j\>2rh
χ(α(|ρ-ϋ-2ΓΛ))(-2)/*||^Γ)||.||»/('·))||.}
.(Ό ι
1/2
1/2
(7.1.27)
Let r = 3,4,···,&. Since for j = 1,···,η, \m \ < 2 and for \t\ <
Γι, |ajr_1)| < α^7-1), and a^~^ is independent of j, considering (7.1.6)
we get that right hand side of (7.1.27) does not exceed
2«('-χ){Σ Σ ι}
J=1 b-j|<2r/i
1/2
1/2
+™{Γ~1}{Έ Σ мь-.л-2гл))<-2>/·}
J=1 \p-j\>2rh
< cc~ll2an[rll\h + l)1/2 + α^μ-ι). (7Л.28)

7.1 Rate of convergence in distribution for a-mixing and p-mixing sequences 177
For r = 2 we proceed in the same way but we estimate m ' for j = 1,2 · · ·, η
according to (7.1.22). One gets that for r = 2 the right hand side of (7.1.27)
does not exceed
cc-1/2d3/*(h + l)b/2t2/a2n + cc-^a^d^ih + φ2/σ2η. (7.1.29)
Adding over r from 2 to /с in (7.1.27) and considering (7.1.28), (7.1.29) and
(7.1.23), we get the proof of (7.1.25).
Finally, since for к > (logn)/(81og2) we have that (l/2)4fc < n"1/2,
(7.1.26) follows from the proof of Lemma 7.1.2, (7.1.6) and (7.1.23). Lemma
7.1.3 is proved.
Lemma 7.1.4. We have
.(D,
Σ \EYjJui | < 32c-1d1/san(a(h + l))^1)/*.
(7.1.30)
// (7.1.23) is satisfied, then
к η r—1
r-l
itz.
(r)
r=2j=l 1=1 1=1
Proof. We only prove the second inequality since the first can be
proved analogously. From the definition of zp , we have that for all
r — 2,3, ···,&, Zj(r' and zfr> cannot be equal to zero simultaneously.
Without loss of generality we assume that i)(r) φ 0 and z^r) φ 0. Then,
according to Lemma 1.2.4
r-l
r-l
»
\e(Yj Π ii°eiteir)) - tffa Π *ί°) Де"4
|я(^П^е^(Г))-£^-П^Ее'
=1
|Wmf
1=1
+
+
г=1
ΜΝ - .ite/">
(г)
i=l
„«£(г)
£eU2'' £eU2J
(r)
£e"zJ
Μ
< 48^/'(а(Л + l))(s-1)/s2r-V^n·

178 Chapter 7 The Berry-Esseen Inequality and the Rate of Weak Convergence
Adding the inequalities obtained over all j = 1,2, · · ·, η and r = 2,3, · · ·, k,
we use (7.1.6) and (7.1.23). Lemma 7.1.4 is proved.
Proof of Theorem 7.1.1.
According to the basic method of the proof , we first establish the
differential equation (7.1.4). Denote a = ££=1 a{k)8^2+6\ let
aQ = cd}lsan{a{h + l)is-2Vs/c0,
ai = cd2'°{l + a)VV„(a(fc + l))^2^/^2,
α2 = cd3/s(h + l)2/c0a„, α3 = cd(h + l)s_1 /'co<~2:,
bo = cdl's{{h + l)1'2 + a^2)(a(h + 1))(*-2)/7<£/2 + a0,
b2 = c<?/°((h + l)1'2 + all2){h + l)2/c0/2a2,
T2 = min{l/a0, l/(6a2), (l/6a3)1/(s-2)}.
Suppose that I < h < η — 1, 2 < к < η — 1 such that
k>^~z, k3l2^k{a{h+l))lls <1, 2kh + Kn. (7.1.31)
8 log 2
From (7.1.3), Lemmas 7.1.1, 7.1.3 and 7.1.4, it follows that
/ή(*) = Η + θα(*))/η(<) + θ6(ί) as |ί|<Γι (7.1.32)
where |θ| < 1,
α(ί) - α0 + αχ|ί| + α2*2 + α3|*Γ~\ b(t) = b0 + Μ2·
Next, we solve the linear differential equation (7.1.32) and get
l/n(0-e-<2/2|
< |*0|е-'2/2+|жо1
,2/о /-1*1 ru2 /Ι*! ί
+ e~* /2 / 6(u) exp^ — + / a(v)dv \du, (7.1.33)
Jo L 2 J\u\ >
where x0 = /0* θα(η)ώ. Let 0 < и < t. Then for a\ < 1/6 and |£| < T2 we
have
/■* t2 — u2
/ a(t7)dt7<l + — , (7.1.34)
Ju 4
and it is easy to see that
r\t\
/ u2 exp(u2/A)du < 2|i| exp(*2/4), (7.1.35)
Jo
r\t\
/ exp(u2/4)du < min(4/|i|, |i|) exp(*2/4). (7.1.36)
Jo

7.1 Rate of convergence in distribution for a-mixing and p-mixing sequences 179
From (7.1.33)-(7.1.36) it follows that for \t\ < min(TbT2) and ax < 1/6
\Ut) - e~^\ < (oolil + °±t> + Щμ|3 + ^|^)β-2/4+ι
16 s
+ еЬ0{щМ} + 2еЬ2Щ. (7.1.37)
By Lemma 1.2.4 we have
σ2 < (1 + 12a)d2/sn,
and hence 1 < (1 + \2afl2K, К = dlls/cQ12. Thus from (7.1.37) and
(7.1.1) we get
(h + iy-1 , ^(h + lf
Ап<с{к°Щ^ + Ю
ση ση
+ K2{{h + l)ll2 + α1/2)^
+ JC2(l + aY/2an(<*(h + l)){s-2)/2s
+ /С((Л + 1)1/2 + α1/2)(α(/ι + 1))(S~2VS} (7.1.38)
where σ2 = аЦс^.
Finally, we prove (7.1.7). Let 0 < λ < λ2 = ^2pl logan and h =
№^logornl, η > 1. Then Щ& < t^U-. Hence, by the condition
[ \δ
λδ — log ση
а(п) < Ke λη, we obtain
,1/2
α1/2=(Σ(^)Ρ+ί))
τ=1
This means
For the chosen /ι,
< 1^/2(2+*)^+ 1)l/2/(log~n)l/2
Δ
<C(K,6)(h + \)ll2. (7.1.39)
1 < C(K, 6)(h + l)x/2/C. (7.1.40)
(α(Λ+ 1))5/2(2+5) < KS/2(2+S)e-\8(h+l)/2(2+S)
< К6/2(2+6)2-2^ (7л 41)
(α(Λ + 1))*/(2+*) < ^/(2+δ)σ~4. (7.1.42)

180 Chapter 7 The Berry-Esseen Inequality and the Rate of Weak Convergence
We get from (7.1.39)-(7.1.42) and (7.1.37) that
An<C(K,S)
<C(k,6)K
■лб
(7.1.43)
k>
It remains to check whether there exists a. k,2 < к < η — 1, such that
[4(2+
log η
for h = д^ ' log ση |, where \\ < λ < Аг we have
k^4k(a(h + 1))χ/(2+ί) < 1, 2kh + 1< n. (7.1.44)
81og2'
It is easy to verify that for all large n(n > no) and for
■(4/6)logan-(logK)/(2 + 6)-
*=[!
(3/2)+log 4
the first two inequalities hold and the third inequality does not constrict
the interval of change of the parameter λ.
Considering (7.1.40), we have
n~l < C{K,6)K?+8{h + l)1+6/G8n.
This also proves Theorem 7.1.1 for all η < щ. The proof of Theorem 7.1.1
is completed.
The proof of Theorem 7.1.2 is analogous to that of Theorem 7.1.1.
We only indicate that for this it is sufficient to set h = [σ£], where a =
2<5/(/3 + l)(o + l),and
к
-ι
(αμ/(2 + δ)) logan - (logК)/(2 + 6h
(3/2)+log 4
in (7.1.31)-(7.1.33).
Remark 7.1.2. Tikhomirov (1980) proved the Berry-Esseen
inequality for the p-mixing sequence with the exponential rate of decay of p(n).
Zuparov (1991) improved his method to prove the following theorem in
which there is a better order of Δη when i£|Xi|s<oo, 2<s<so(<
1 + л/3), and the rate of p(-) was weakened.
Theorem 7.1.3. Let {Xn,n > 1} be a strictly stationary p-mixing
sequence with ΕΧχ = 0 and
p(n) < Kn~\ θ > 0, К > 0,Е\Хг\3 < oo,

7.2 The rate of weak convergence for a ^-mixing sequence
181
,„ 0-1 //0-l\2 4 + 20
2<3<8ο(θ) = — + ή( — ) +-§-.
If σ\ — ES\ > τηΕΧ\. Then there exists a constant С = С(з,в,К,т)
depending only on s,0, К and τ such that
Δη<0(8,θ,Κ,τ)β3/η^-2ν\
where β8 = Е\Хг\3/(ЕХ2у/2, s0 < 1 + y/3.
The proof of Theorem 7.1.3 will not be presented here.
7.2 The rate of weak convergence for a (^-mixing sequence
Let {Xn,n > 1} be a sequence of random variables with EXn =
0, σ\ = ES2. Define the partial sum process as follows:
.2 _ ji
(0 = -J-(S[nt] + Γ" ^J X[nt]+i), (7.2.1)
^n σ[ηί] + 1 σ[ηί]
Χ <>[*]£*<% <°[nt] + V
Denote by Pn the distribution of Wn in C[0,1], i.e. Pn = Ρ ο W'1.
The weak invariance principle holds, i.e. Pn =$> W, that is equivalent to
L(Pn,W)—0, (7.2.2)
where L(P, Q) is the Levy-Prohorov distance:
L(P, Q) = ίηί{ε :VBG^, P(B) < Q(B£)+e, Q(B) < Ρ(Ρε)+ε}, (7.2.3)
where Τ is the Borel σ-field of C[0,1], Pe is the ε-neighborhood of Borel
set P,
B£ = {y:yeC[0,l],3zeB,\\y-z\\<e}, \\y-z\\= sup |ι/(ί)-*(ί)Ι·
o<t<l
Denote
Λη(ε) - sup \P(Wn G B) - P(W G Be)|,
It is obvious that
L(Pn, W)=inf(eVAn(e)). (7.2.4)
Then without changing the distribution of {Wn,n > 1}, we can redefine
process {Wn,n > 1} on a richer probability space together with the
standard Wiener process {W(t),t > 0} such that
Χ(ε)<Ρ{||^η-^||>ε}.

182 Chapter 7 The Berry-Esseen Inequality and the Rate of Weak Convergence
Therefore, in order to estimate the rate of weak convergence, we need only
consider the following inequality:
L(Pn, W) < inf(e V P{\\Wn - W\\ > ε}).
For the independent random variables {Xn,n > 1}, Prohorov (1956)
gave a precise estimation :
L(Pn,W) = 0{L\'A log2 L3),
where L3 = Σ1=ι ЩХк\3/<г1- For i.i.d.r.v.'s, Borovkov (1973) proved: if
£|Χ!|2+δ
< oo, 0 < δ < 1, then
L(Pn,W) = 0{ΐ2$+6)) = O(n-I^I). (7.2.5)
Borovkov (1973) also pointed out that the weak convergence rate (7.2.5)
can not be improved if we use only the Skorohod method. Utev (1981)
generalized this estimation to the case 0 < δ < 3 and gave
L(Pn,W) = 0(n~^) ),when£|Xi|2+* < oo, 0 < δ < 3. (7.2.6)
Utev (1984) gave the same estimation as (7.2.6) for <£>-mixing sequence
under a stronger condition on the rate of decay of φ{η).
Theorem 7.2.1. Let {Xn,n > 1} be a strictly stationary φ-mixing
sequence of random variables with EX\ — 0,EXj — 1 and Ε\Χι\2+δ < oo
for some 0 < δ < 3. Suppose
φ(η) < cri~g
where g > j(u)(j(u) - 1), и = (12 + 5<5)/(2(3 - δ)), j(u) = min{2fc : 2k >
u,k G N}. Then
L(Pn,W) < сп~ТОУ.
By using Lemma 2.2.10, Lu (1993) weakened the conditions and proved
the following theorem.
Theorem 7.2.2. Let {Xn,n > 1} be a φ-mixing sequence of random
variables with EXn = 0, A0 — supnEX2 < oo, As = supn Ε\Χη\2+δ <
oo, 0 < δ < 3. Suppose that
(i) if σ\ —> oo as η —> oo;

7.2 The rate of weak convergence for a ^-mixing sequence
183
(it) φ{η) < спГд, g > (3<5 + ε)/(2(3 - δ)) V (2 + ε) for any ε > 0.
Then we have
L(Pn, W) = 0(n~^)). (7.2.7)
Proof. By using (i), A$ < oo, Lemma 2.2.2 and Lemma 2.2.10, we
have
Ε max \Sk(i)\2+6 < cUrifSr + Ε max |X;|2+*)
1<г<п V k<i<k+n '
< cn1+8'2. (7.2.8)
Without loss of generality, we can assume that σ\ = con, cq — 1. Then Wn
is a random polygonal line with nodes at (fc/n, Sfc/д/п), к — 0,1, · · · , η.
Denote
Х}г) = Xi/dX-l < yVn) ~ EXiI(\Xi\ < yy/n) i = 1, · · ·, n,
where у = η,-^2(Σ^=ι E\x.\2+sy/(2+s) < п~щ&щД&*ш Note that there
exist all finite moments of X\ for any given n.
Let Sfc — Σ?=ι -^Ч and Wn be a random polygonal line with nodes
at(k/n,SP/y/n). By using Lemma 2.2.10 , we have
p{\\wW-Wn\\>cn-sM3+V}
< P{ max |4=(5fc - 5^)1 > cn"5/2^)}
< c„-3(2+*)/2(3+i)E max |5fc_5W|2+i
< m-3<2+W+*)[( ma^lS* - S<x)|2)
+ £вд-Х<1>|2+*]
< cn-*M3+s\ (7.2.9)
2+6
2
Next, put / = [n3/(3+*)-r>], m = [ηδΚ3+δ)+% η > 0 (specified later
on), write η — Im + r, 0 < r < /. Let Wn be a random polygonal line

184 Chapter 7 The Berry-Esseen Inequality and the Rate of Weak Convergence
with nodes at (kl/n, S^ /y/n), к = 0,1, · · · , m and (1, Sn /y/n). We have
P{\\w^-w^\\>b}
m
к=0 -Ъ-
m
< сУЬ-ип~и12Е max ^(г)!"
— t_j Ki<l' ftt
fc=0
< cmb~un~ul2lul2
where we have applied Lemma 2.2.10. If we take b — en <5/2(3+<5)j u >
2 + 36/[(3 + 6)ч], then
(y) <™ 2<3+*).
I
Therefore we have
^{||^12) - W^ll > cn~^) } < сгГ^У. (7.2.10)
Put /ι = [ηθ], θ > 0 (specified later on). Let d — l — h^
Sfc - 2-f (fc-l)M-i'
Wn be a random polygonal line with nodes at (к1/п,^=1 Q /y/n), we
have
= ρ{,<Κ.,ΐ5«)-ΣΣ^„1+ίΐϊ^}
к h
= p{ max | У У X^, v . ,. · | > by/n\
- - j=l г=1
*,*-»-/"{[ ι<Κ Ε(ΣΣ|Χ«1)/+,+ί|)2]
- - jf=l г=1
m+1 h

7.2 The rate of weak convergence for a (^-mixing sequence
185
where the first term does not exceed c(mh)ul2 and the last sum does not
exceed cmhul2 — o((mh)ul2) by Lemma 2.2.4, therefore
P{\\W^-W^\\>b}
< cb-un-ul2{mh)ul2 < cfTu(/i/Z)u/2.
If we take b = cn~e^3+^ and
then {h/l)ul2 < cn-e(»+D/2(3+6)) so that
P{\\WW - W^W > cn-s№+V} < cn-s^3+s\ (7.2.12)
In the remain of the proof, imitating the proof of Utev (1984), i.e. by
Berkes and Philipp (1979) Theorem 2, we have
P{\\WW - W™\\ > ap(h)n/l} < c<p(h)n/l, (7.2.13)
where Wn is the random polygonal line with nodes (kl/n, J^=i £j /V™)-»
к — 0, 1, · · ·, m + 1, the Q ' are independent and distributed same as Q \
And by Sakhanenko (1981), we have
P{\\W^ - Wi4)|| > q^1} < цФ1 и>2, (7.2.14)
where Wn is the random polygonal line with nodes (Μ/η,Σ^=1 Yj/л/п),
к = 0,1,···,7тг + 1, the Yj are independent normally distributed random
variables with EYj — 0, Varlj — VarC· \j = 1,2, · · ·, m + 1 and
m+l m+1 /0
г=1 г=1
It is easy to see that by (7.2.8) we have
m
qu = сп-и'2 £^<Х)Г < cn-"/2m/"/2
г=1
< φι//)1""/2, (7.2.15)
so that
when u > 2 + 36/(3 + δ)η.

186 Chapter 7 The Berry-Esseen Inequality and the Rate of Weak Convergence
At last, if we take
л 2(3-<5)/ 36 \
then
φ{Κ)η/1 = 0(η~9θ+^+η) < αη-δΙ2^δ\ (7.2.16)
Combining with (7.2.11), we have
+ η) < θ <
2(3-g)/ ЗД ь 3-{ 477
(3 + ε)δ V2(3 + δ) η' - 3 + 6 3'
that is to say, we need take η as follows:
3-5 3εδ
0 <η <
3 + <5 2(9 +3<5+ 2ε<5)
Combining (7.2.9)-(7.2.16) together we obtain
p{\\Wn _ Щ > ^-«/2(3+*)} < ^-«/2(3+*).
Since L(Wn, W) < infe(e + P{||Wn - W\\ > ε}), (7.2.7) holds true. The
proof of Theorem 7.2.2 is completed.
Remark 7.2.1. Utev (1984) discussed the rate of weak convergence
for an α-mixing sequence and obtained the following result:
Let {ХП5 п > 1} be a strictly stationary α-mixing sequence with EX\ —
0, EX\ = 1 and £|Xi|2+* < oo, 0 < 6 < 3. Assume that
0 < σ2 := EX\ + 2 ]T ΕΧλΧη+1 < oo
n=l
a{n) < cn~9, n — 1,2, · · ·
n=l
and
where
5 > max^rVMUH + ^),«ГЪ'(2(2 + 6))(j(2(2 + 6)) + 5X)),
12 + 5<5\ с δ
и — max
(^>-й^)- *■-
2(3-£)>" х (2 + 6)(3 + 5)'
j(u) is defined as in Theorem 7.2.1. Then
L(Pn,W)<cAsn-s№3+s»,

7.2 The rate of weak convergence for a <£>-mixing sequence
187
where с = с(А,д,6).
If δ = 1, i.e., assume that Ε\Χχ\3 < oo, a(n) = 0(η~(438+ε)), then
L(P„,W) = 0(n-V8).
Gorodezkii (1983) has also given a similar result.
Remark 7.2.2. Yoshihara (1979) discussed the rate of the weakly
invariance principle for the strictly stationary absolutely regular sequence
{Xn,n > 1}, and proved that if EXX = 0, Ε\Χλ\Α+δ < oo for some δ > 0
and Σ™=ι п/3(п)*/(4+*) < oo, then
L(Pn,W) = 0(n-1/8(logn)1/2).

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Part III Almost Sure Convergence
and Strong Approximations
In this part, we study the almost sure convergence and the strong
approximations of the partial sums of a mixing dependent sequence. Since
1960s, the almost sure convergence has been discussed by some authors,
e.g., Iosifescu and Theodorescu (1969) obtained the 0-1 law, the strong
law of large numbers and the convergence of random series of a (^-mixing
sequence, etc. The complete convergence of various mixing sequences has
been studied deeply by Shao et al. in the past ten years, from which the
elegant results of the strong laws of large numbers follow. We shall discuss
these in Chapter 8.
The strong approximations of the partial sums Sn = Y%=\ Xk for a
mixing sequence {Xn,n > 1} by a Wiener process were done by Philipp
and Stout (1975) et al., and were improved comprehensively by Shao and
Lu; the limiting behaviour of the increments of partial sums for a
mixing sequence was obtained by Lin et al. Those elegant theorems will be
discussed in Chapter 9 and Chapter 10 respectively.
The strong approximation of a mixing sequence with set-indexed will
be introduced in Chapter 11.

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Chapter 8 Laws of Large Numbers
and Complete Convergence
We shall introduce the Borel-Cantelli lemma, the weak law of large
numbers and the strong law of large numbers for a <£>-mixing sequence
in the first two sections. Since the concept of complete convergence was
raised by Hsu and Robbins (1947), this subject has attracted the attention
of many mathematicians. The complete convergence of weakly dependent
sequences has been obtained by some mathematicians. We shall introduce
the complete convergence of a <£>-mixing sequence, a p-mixing sequence and
an α-mixing sequence in Sections 8.3-8.5 respectively. At last, we discuss
three problems which are posed by Prohorov for the complete convergence
of /o-mixing sequence in Section 8.6.
8.1 Weak law of large numbers
Theorem 10.1.1 of Chow and Teicher (1978) gave a sufficient and
necessary condition for the weak law of la'rge numbers for an array of
independent random variables {Xnj, 1 < j < kn}. Du (1993) generalized this
result to the <£>-mixing case and proved the following theorem. Denote
Theorem 8.1.1. Let {Xnj-, 1 < J < kn —> oo} be a random variable
array which is φ-mixing in each row. There exists an integer Μ such that
φ(Μ) < 1/2. Then for some real numbers An
Sn-An-^0, (8.1.1)
P\ max \Xnj\ > ε] -► 0 for any ε > 0, (8.1.2)

192
Chapter 8 Laws of Large Numbers and Comolete Convergence
as η —> oo iff
ten
Σ P{\Xnj\ > ε} -► 0 for any ε > 0, (8.1.3)
Σ
j=i
Var(£;XnjI(\Xu3\ < 1)) - 0, (8.1.4)
in which we can take
An = f^EXnjI(\Xnj\ < 1) + o(l).
3 = 1
The proof of Theorem 8.1.1 will needs the following lemmas.
Lemma 8.1.1. Let {Xn,n > 1} be a φ-mixing sequence. Denote by
m(Y) the median of a random variable Υ. Then we have
P{ max |5,· - m(Sj - 5n+fc_x + Sj(k - 1))| > ε}
* rhcpi{k -1} i<3^k-ilXjl + |5-+fc-11 *ε) (8Л-5)
/or large к and some C, 0 < С < 1/2, where Sj(k) = ^=J-+1 Xt*·
Proof. Take A: so large that С :— <£>(&) < 1/2. Let
fmin {
l<j<n
n+1,
mm {j : 5,- - m(Sj - 5n+fc_i + Sj(k - 1)) > ε},
τ : -
if the above set is empty.
Denote
Bj = i^i "" ^n+fc-i + Sj(k - 1)
< miSj - 5n+fc_x + S^fc - 1))}, 1 < j < n.
It is clear that P{Bj) > 1/2. Since {T = j} £ Τ{ = σ{Χχ, · · · ,Xj},Bj G
J^-1, we have
(J {B3 f|(T = j)) С {5n+fc_! - £,-(* - 1) > ε}
c{(fc-l) max |Xi| + 5n+fc_1>£}.

8.1 Weak law of large numbers
193
From the (^-mixing property it follows that
P{(k ~ 1), . max , \Xj\ + Sn+k-i > 4
*■ l<j<n+k—l >
>J2P{Bj,T = j}
3 = 1
>Σ{Ρ{Β3)-φ{1τ))Ρ{Τ = 3}
>{\-C)P{l<T<n}
= {\-с)р{Шп{з3
- m{Sj - Sn+k-i + Sj(k - 1)) > ε}. (8.1.6)
By the same way, we have another inequality when Xj is replaced by —Xj.
(8.1.5) is proved.
Denote
B = (max \S,(n-l)\ > ε),
Bj ~ ι \Sj(n ~ J)\ > ε5 max \Sk(n — k)\ < ε >, 0 < j < n,
η
Bn = {\Xn\ > ε}, ^- = ЦД, 0<j<n Fn+1=0.
Lemma 8.1.2. Letf {Xn,n > 1} be α φ-тгхгпд sequence with \Xj\ <
r < oo, j — 1, 2, · · ·, n, and φ{Μ) < 1/6 /or some Μ > 0. ГЛеп
Pi m > (1 ~ M^)) ^{™Χ!<,<η Sf} - 4ε2
W - 3£?{maxi</<n Sf} + 5M2r2 + 12(e + r)2 - 2ε2' ^ " " ;
Proof. For 1< j < fc < η we have
max \Si\ < \Sk\ + max \Si(k - t)| < 2 max \Si(k - t)|. (8.1.8)
1<г<& 0<г<& 0<г<к
If 0 < Μ < η, 0 < г < η - Μ, we have
and
max |Sj| < max \Si\ + Mr, г < j < г + M, (8.1.9)
max |5/| < |5<| + Mr+ max |S<+Af(i - i - M)\ (8.1.10)
z+M</<j i+M<l<j

194
Chapter 8 Laws of Large Numbers and Comolete Convergence
for г + Μ < I < j < η. From (8.1.8)-(8.1.10) it follows that
max |5/| = maxi max ISJ, max 15/1 >
i<i<j Ικκΐ+м1 fc"t+M<z<j' '/
< max \Sk\ + Mr + max \S{+m(1 - i - M)\
l<k<i i+M<l<j
< max \Sk\ + Mr + 2 max . \St(j - l)\
1<к<г i+M<l<j
< max \Sk\ + Mr + 4 max |S/(n - /)|
1<&<г i+M<i<n
for j — 1,2, · · ·, n. Therefore for any given г, 1<г<п — Mwe have
max \Si\ < max \Sk\ + Mr + 4 max |S/(n - /)|. (8.1.11)
1<ί<η l<k<i i+M<l<n
Note that
max \Si(n-l)\IBj
j<l<n
< \Xj+x\IBj + . max |S,(n - Z)|/Bj < (r + e)/Bj
j + l<.i<.n
for j = 1,2,···, η and (a + b + c)2 < 3(a2+ b2+ c2). By (8.1.11) we obtain
Ε max Si2/β = Y^ Ε max 5ι2/β.
1</<п l 4^ 1</<п Z J
Μ η
<V# max S?IB.+ Υ e{ max \Sk\
j=0 j=M+l - ~J
Ί 2
+ Mr + 4 max |S/(n - Ш /β.
j<l<n J J
Μ η
< У" Ε max S?IB. +3 V £ max 5?/β.
+ (3M2r2 + 12(r + ε)2)Ρ(5). (8.1.12)
Put Yj — max!<fc<j-M |^|, Μ < j < η. By Lemma 1.2.10 and noting

8.1 Weak law of large numbers
195
that P{Fi) is non-increasing for г, we get
η
Υ* Ε max SllBi
= Σ ΕΨ*
j=M+l
= Σ {EYflFj-EYflFj+1)
j=M+l
= EY^+1IFm+1+ £ E{Yf-Yf_x)IF.
j=M+2
<EY*I+1{P(FM+1) + 2ip(M)}
+ £ E(Y?-Y?_1){P(Fj) + 2<p(M)}
j=M+2
<EYZ{P(FM+i) + 2<p(M)}.
Moreover
Μ
^EmaxjflB, < {Mr + e)2P{B)
j=o
<2(M2r2 + s2)P(S).
Inserting these into (8.1.12) we have
Ε max S?Ir
l<l<n L
< ЪЕ max Sf(P(B) + 2φ(Μ))
1<1<η
+ (5M2r2 + 12(r + ε)2 + 2ε2)Ρ(5). (8.1.13)
On the other hand
Ε max SflB = Ε max S? - Ε max S?Ibc
l<l<n L l<l<n L l<l<n
> Ε max Sf - 4ε2(1 - P(B)). (8.1.14)
l<l<n
Combining (8.1.14) with (8.1.13) yields (8.1.7).
Lemma 8.1.3. Let {Xn,n > 1} be as in Lemma 8.1.2. If Sn almost
surely converge to a random variable as η —> oo, then £тахккп S| are
convergent.

196
Chapter 8 Laws of Large Numbers and Comolete Convergence
Proof. Denote £>n,n+m - {S£™{\Si(n+m-i)\ > ε}. If φ{Μ) < 1/6,
by Lemma 8.1.2
■* \J-Sn,n+mj
> (l-6^(M))£{maxn</<n+m5/2}-4£2
3£{maxn<i<n+m Sf} + 5M2r2 + 12(e + r)2 - 2ε2'
Using the reduction to absurdity, if £?maxi<fc<n S| are divergent, then for
any given η > 1
Ε max Sf > Ε max Sf - Ε max Sf > δ0 > 0 (8.1.15)
7l<Z<7l+m 1</<П+т 1<Z<71
о о
for large 77г. On the other hand, from Sn —^ 5 we have
P{Dn,n+m} = P\ max |5»(n + m - г)| > ε}
^п<г<п+т У
<Р{ max |Sn(i)| > εβ\
+ Ρ{|5η(τη)|>ε/2}-^0,
as η, m —> oo.The contradiction to (8.1.15) implies that Lemma 8.1.3 holds
true.
Lemma 8.1.4. Let В — <|maxi<j<n \Xj\ > ε>,Τη = ^™=1/(|Хг| > ε).
If φ{Μ) < 1/2, we have
> (1-2Ψ(Μ))ΕΤ
v } ~ ETn + Μ +1 v y
Proof. Denote
T0 = 0, Tn(m) = Tn+m - Tn,
Bn = {\Xn\>e},
Bj — { max \X{\ < ε, |Xj| > ε},
j<i<n
η
Fj = [jBi,j = lt---tnt Fn+1 = 0.

8.1 Weak law of large numbers
197
We have
Μ
ETnIB = Σ ЕЩ-г + Tj-^n - j + 1))ВД)
η
+ Σ Ε(Τ^μ-ι+Τ^μ-ι(Μ)
j=M+l
+ Tj-1(n-j-l))I(Bj)
Μ Μ
<£МР(В,-) + £Р(Я,·)
+ Σ (ΕΤ^Μ+1Ι(Βι) + ΜΡ(Βι) + Ρ(Β3))
j=M+l
< JT ΕΤ3-Μ-ιΙ(Β3) + (Μ + 1)Ρ(Β).
j=M+l
By Lemma 1.2.10
Σ Ε(Τί-Μ-ιΙ(Βά))
j=M+l
= Σ Я(Г^м-1(/(^)--ад+1)))
η
= ET^Fm+2) + Σ #/(|*;-Μ-ι|>ε)/(^·)
j=M+3
η—Μ
< { Σ £?/(|Χ<| > ε)}{Ρ(ΡΜ+ι) + 2<^(Μ)}
г=1
< ΕΤη{Ρ{Β) + 2<^(Μ)}.
Therefore
#τη - #τη/(β)
< ΕΤη{Ρ(Β) + 2^(Μ)} + (Μ + l)P(S),
as desired.
Proof of Theorem 8.1.1.
The part of "г/". It is clear that (8.1.3) implies (8.1.2). In order to
show (8.1.1), denote
ten
3 = 1

198
Chapter 8 Laws of Large Numbers and Comolete Convergence
By (8.1.4) we have Vn - EVn -^ 0. From (8.1.3) it follows that
к
P{Vn φ Sn) < Σ P{\Xnj\ > 1} = o(l), (8.1.17)
j=i
i.e., Sn - EVn -£♦ 0. (8.1.1) holds true for An = EVn + o(l).
The part of "only if". From Lemma 8.1.4 we have
p/ ,γ .. ^(l-2V(M))EEkj=1I(\Xn3\>e)
F< max \ХпЛ > ε> > г ,
so that (8.1.2) implies (8.1.3). In order to show (8.1.4), let Y^ be an
independent copy of Ynj and put
ten
у*. — у—у1, v* — Y^ У*
3 = 1
From (8.1.17) and (8.1.1) it follows that Vn-An -^ 0. Therefore V* -^ 0.
Denote V*k = Zj=i Υ£· By Lemma 8.1.1
- o(l). (8.1.18)
Hence by (8.1.2) and (8.1.18)
+ Σ Ρ{\Υ:3\>ε/Μ} = ο(1).
j=kn-M+l
Using the same method as in the proof of Lemma 8.1.3 we get that VarV^
tends to 0. Thus VarV^ = VarV^/2 approaches 0 as well. Theorem 8.1.1
is proved.

8.2 Strong laws of large numbers
199
8.2 Strong laws of large numbers
First we show the Borel-Cantelli lemma for a <£>-mixing sequence.
Theorem 8.2.1. Let {Xn,n > 1} be a φ-mixing sequence and let
{Tn = σ(Χη),η > 1} be a sequence of σ-fields. Then for any given
An G Tn, P{An, i.o.} = 0iff
oo
Σ,Ρ{Αη}<οο. (8.2.1)
n=l
Furthermore, that Σ™=ι P{An} — oo implies P{An, i.o.} = 1.
Proof. For the first part, we need only to show that P{An, i.o.} =
0 implies Σ™=1Ρ{Αη) < oo. Otherwise, if Σ™=ιΡ{Αη) = oo. By the
assumption, there exists an / > 0 such that δ :— φ{ί) < 1, and there is
some j, 0 < j < / — 1 such that Σ™=ι P{Ani+j} = oo. We have
η
p{ (J All+j} =P{Anl+j} + P{Acnl+j П A{n_1)l+J} + ■■■
i=m
+ P{Ani+j П · · · П A\m+X)l+- П Aml+j}.
By the y-mixing property we get
P{\jAil+j]
i=m
> P{Anl+J} + P{A{n_1)l+j}(P{Acnl+j} -«) + ..·
+ P{Aml+J}(P{Ac{m+1)l+j η · · · П Acnl+j} - δ)
>J2P{AU+3}(P{( (j Ail+J)C}-6).
i=m i=m+l
From EkLiP{Aki+j} - oo it follows that p{([JZmAi+j)C} < ^ i.e.,
PJ|Jgm Л;/+Л > 1 - <5, therefore P{An, i.o.} > 1 - <5, which contradict
with P{An, i.o.} = 0.
Secondly, if Σ™=ι Ρ{Αη} — oo, it follows from the above discussion
that
PJlimsupAn\ = P{An, i.o.} ф 0.
It is clear that lim sup An G Γ£°=ι V£n ^· We now show that (XLi V£n ?χ
is a trivial σ-field. Otherwise, there exists a set β G H^Li VSn^*b 0 <
P(5) < 1. By the (^-mixing property, for every Л G \/Γ=ι ^г we have
|P(AB) - P(A)P(B)\ < ηΡ(Α), (8.2.2)

200
Chapter 8 Laws of Large Numbers and Comolete Convergence
where 77 < 1/2. It is easy to check that the class of sets which satisfies
(8.2.2) includes the σ-field V^i ?и so that taking A — В we obtain
P(B) - P{B)2 < ηΡ(Β),
which implies P(B) > 1/2. On the other hand, we have also Bc £
H^Li VSn^· A contradiction leads P{An, i.o.} — 1 as desired.
Corollary 8.2.1. Let {Xn,n > 1} be a φ-mixing sequence of
identically distributed random variables. Put Sn = Σ£=1 X-k- If Sn/n —> b a.s.,
where b is a finite constant, then E\X\\ < 00.
Proof. From Sn/n —> b (a.s.) we have
■sin bn bn_i П — 1
= · ► 0 a.s.
η η η — 1 η
Therefore P{\Xn/n\ > ε, i.o.} = 0 for any ε > 0. By Theorem 8.2.1
00
Е\Хг\ < ^Р{\Хг\ > η} < οο.
η=0
For the (^-mixing sequence of identically distributed random variables,
we have the following Marcinkiewicz strong law of large numbers.
Theorem 8.2.2. Let {Xn,n > 1} be a φ-mixing (p-mixing) sequence
of identically distributed random variables with
00 00
£ <//2(2") < 00 (Σ P(2n) < «О.ВДГ < oo
n=l n=l
for some 1 < r < 2. Then
- JT(Xi - ЕХ{) = о(п'^'г^) a.s. (8.2.3)
τι .
г=1
Theorem 8.2.2 is an immediate consequence of Corollary 8.3.4
(Corollary 8.4.2), so we omit its proof here.
Corollary 8.2.2. Let {Xn,n > 1} be a φ-mixing sequence of
identically distributed random variables with Σ™=ι ^1/'2(2n) < 00. Then Sn/n —>
b a.s. iff Ε\Χχ\ < 00 and b — EX\.

8.3 Complete convergence for <£>-mixing sequences
201
Remark 8.2.1. Xue (1994) showed that Theorem 8.2.2 is also true
for a (^-mixing sequence {Xn5^ > 1} with Σφιί2{2η) < oo, if there exists
a random variable X such that for any χ > 0, P{|Xn| > x} < P{\X\ > x}
and E\X\r < oo for some 1 < r < 2.
Remark 8.2.2. Iosifescu and Theodorescu (1969) have discussed the
convergence of a series of a (^-mixing sequence, for example, they showed
that for Σ Χη a.s. convergence is equivalent to convergence in probability
under some condition. They have also discussed the following strong law of
large numbers: if an | oo, Σ™=ι VarXn/a^ < oo and Σ™=ι φ1^2^) < oo
then
— (Sn-ESn)-^0 a.s.
Remark 8.2.3. Chen and Wu (1989) have proved a strong law of
large numbers for an α-mixing sequence. Suppose that s\ipnE\Xn\p < oo
for some ρ > 1, and
a(n) = I
0(n 2p-2 e) if 1 < ρ < 2,
0(η~ρ~ε) if p>2.
Then (Sn - ESn)/n = o(l) a.s.
8.3 Complete convergence for (^-mixing sequences
The concept of complete convergence was introduced by Hsu and Rob-
bins (1947). They showed: if {Xn,n > 1} is a sequence of i.i.d. random
variables with EX\ — 0, EX\ < oo, then
J2Pi\Sn\ >en} <oo
71=1
for any ε > 0. Baum and Katz (1965) showed that EXX = 0, Е\Хг\гг < oo
for r > 1, 1 < t < 2 iff
oo
Σ nr-2P{\Sn\> en1/'} <oo
n=l
for any ε > 0. Bai and Su (1985) obtained:

202
Chapter 8 Laws of Large Numbers and Comolete Convergence
Theorem 8.3.1. Let {Xn,n > 1} be a sequence of i.i.d. random
variables and r > 1, 0 < 2 < 2, h(x) be a slowly varying function as χ —► oo.
Then the following conditions are equivalent:
(i) Е\Хг\нН{\Хг^) < oo,
(ii) EST=i nr-2h(n)P{\Sn - nb\ > en1'1} < oo for any ε > 0,
(™) Σ™=ι nr~2h(n)p{™pk>n {Sk-kbl/k1/1 > ε} < oo for any ε > 0,
where b = EXX if 1 < t < 2, and 0 if 0 < t < 1.
Since 1970s, the complete convergence of a mixing sequence has been
discussed by some research works. The best result corresponding to that
for an i.i.d. sequence is due to Shao (1988a).
Let l{x) and β(χ) be positive even functions such that
( l(x), β(χ)/χθ (somefl > 0) and χ2/β(χ)
\ are monotonically nondecreasing
for χ > 0. Shao (1988a) proved the following theorems.
(8.3.1)
Theorem 8.3.2. Let a(x) — Ίηνβ(χ), {Xn,n > 1} be a φ-mixing
sequence of identically distributed random variables with EX\ = 0. Suppose
that
Εβ(Χ1)1(β(Χ1)) < oo. (8.3.2)
// one of the following conditions is satisfied:
α) β(χ)1(β(χ))/χ t? β(χ)1(β(χ))/χ2 I an& there exists a r > 2 such
that
oo 1
V -r-τ-τ < oo (8.3.3)
^-i nlr(n) v ;
n=l ч '
and
[bgn]
I>1/2(2l)<^logKn); (8.3.4)
i=l
b) β(χ)1(β(χ))/χ Τ, β(χ)1(β(χ))/χ2 Ι,
]=п
£Ш-°Ф - — <8·3·5)
^У1/2(Л<оо; (8.3.6)
г=1
с) β(χ)1(β(χ))/χ2 | and there exist qi > q<i > 2 such that
β(χ)1(β(χ))/χ<>> |,

8.3 Complete convergence for (^-mixing sequences
203
oo ι/ \ ι/ο [fogn]
Σ?(^Γ^ΣΛ·)}<« (8-3.7)
n=l ^ ' i=l
Then
°° Kn)
Σ —P{ max P(Si) > εη} < oo, for any ε > 0. (8.3.8)
1 П
n=l
Theorem 8.3.3. Suppose that l(x) is strictly monotone. Let {Xn,n >
1} be a φ-mixing sequence with a common distribution. If (8.3.8) is
satisfied then (8.3.2) holds and
71=1
Σ*(η)-«[η/2])ρ|8 g0gi)>£|<00, forany £>0. (8.3.9)
Remark 8.3.1. If l(n) = 0(l(n) - /([ra/2])), (8.3.8) is equivalent to
(8.3.9).
Let l(n) = nr~1,r > l,/3(n) = n',1 < ί < 2 in Theorems 8.3.2 and
8.3.3, we have
Corollary 8.3.1. Let {Xnin > 1} be a φ-mixing sequence with a
common distribution. The following results are equivalent:
(г) Е\Хг\гЬ <oo, EXi = 0;
(™) Y^=\ nr~2PJmaxi<i<n \S{\ > εη1/* | < oo, for any ε > 0;
(Hi) Σ~=ι пг~2Р{ыРк>п Ш/к1^ > ε} < oo, for any ε > 0.
Choosing l(n) = logn, /3(n) = η*, 1 < t < 2, we have
Corollary 8.3.2. Let {Xn,n > 1} be a φ-mixing sequence with a
common distribution and φ(η) < —(logn)-2 for large n. Then the following
results are equivalent:
(i) E\X1\t\og(l + \Xi\)<oo, ЕХг=0;
(ii) ЕгГ=1^^{таХ1<г<п|5г| > ЕП1'1} < OO, for any 6 > 0.
Corollary 8.3.3. Let {Xn,n > 1} be a φ-mixing sequence with a
common distribution andEXx = 0, Е\Хг\* <ool<t<2. 7/Σ£°=ι <£1/2(2η) <
oo then
oo
Σ
71=1
-P{ max ISA > enllt\ < oo
П U<t<n J

204
Chapter 8 Laws of Large Numbers and Comolete Convergence
for any ε > 0.
Remark 8.3.2. Corollary 8.3.1 can be strengthened as follows: the
same result as Theorem 8.3.1 holds true for a (^-mixing sequence with a
common distribution, i.e., if h(x) > 0 is a slowly varying function, the
following are equivalent:
(i)' Β|ΧιΓ*Λ(|Χι|*) < oo, EXi = 0;
(ii)' Σ™=ι nr-2h(n)P{m3xi<i<n \Si\ > εη1/*} < oo, for any ε > 0;
(ш)' ΣΓ-ι nr-2/i(n)P{maxfc>n {Skl/k1/1 > ε} < oo, for any ε > 0.
where r > 1, 1 < t < 2.
Corollary 8.3.2 can be strengthened similarly.
Proof of Theorem 8.3.2.
It is easy to see that (8.3.1) implies
Put
We have
a(x)/x1/2 t, α(χ)/χ1/θ [ . (8.3.10)
Xni = XJ{\Xi\ < a(rc)}, Sni = ^2Xnj.
j = l
oo
y^p(maX/3(5i)>£n)
^^ί Ti ^ i<n J
n=l ~
l{n)
<yl^±p{m^\Xi\>a{n)\
n=l —
+ V" -^PJmax \Sni\ > α(εη)\
^—^ П *< i<n )
n=l
=:h+I2. (8.3.11)
It is clear that
Ii<Y^l(n)P{\Xi\>a(n)}
n=l
oo
n=l j=n
oo
< Εβ(Χι)1(β(Χι)) < oo. (8.3.12)

8.3 Complete convergence for «^-mixing sequences
205
Now we estimate I2. Without loss of generality we can assume that 0 <
ε < 1/2. By (8.3.10)
n|£Xi/{|Xi| < a(n)}\ < nE|A"i|/{|Xi| > a(n)}
< ^Εβ(Χ1)1(β(Χ1))Ι{\Χι\ > a(n)}
< ^^Εβ(Χ1)1{β(Χ1))Ι{\Χ1\ > α(η)}.
Noting that Z(l) > 0, limn^oo Εβ(Χ1)1(β(Χ1))Ι{\Χ1\ > a(n)} = 0, we
have
n\EXiI{\Xi\ < a(n)}\ < -α(εη)
Δι
for large n. Therefore
h < с f] ^PJmax \Sni - ESni\ > Ια(εη)}. (8.3.13)
n=l ~~
By Lemma 2.2.2 and Lemma 2.2.10, for given q > 2 (specified later on),
there exists a, Cq = C(q) depending only on q such that
P\ max \Sni - ESni\ > -α(εη)\
у- 1<г<п 2 '
[log?
i=l
<Cq(a(en))-o((nexp(3 £ φ1'2^))
.ЕХ11{\Хг\<а{п)}У
+ nE\Xl\qI{\X1\ < α(η)}). (8.3.14)
If condition a) is satisfied, let q = 28(r + 1). From (8.3.10), (8.3.13)
and (8.3.14) we have
n=l ^ ' i=l
71 = 1 ^ '
=: /3 + /4· (8.3.15)

206
Chapter 8 Laws of Large Numbers and Comolete Convergence
By (8.3.4)
OO
η=ι nl(n) 2
OO
<<Σ
-2--1
П=1П1(П) 28
OO *
η=1 ч у
Since /3(x)/(/3(x))/x2 [ implies χ/(χ)/α2(χ) |, it follows from (8.3.10) that
Z^ ^σ/^Λ Ζ^
?-2
2
t^cfl{y) %<*\з) ач-*{э)эФ
η«/2ί(η)
OO -ι
a*(ra) ^ ?V2
4 ' τ =77. ^
"nl(n)
3=П "
= θ№). (8.3.17)
Therefore
OO i/ л П
n=l ^ ' j=0
OO OO ?/ \
j=0 n=j ^ '
< c£/3(Xi)J(/3(Xi)) < oo. (8.3.18)
From (8.3.15), (8.3.16) and (8.3.18) it follows that I2 < oo, i.e., Theorem
8.3.2 holds true under condition a).
When condition b) is satisfied, let q = 2. By the similar discussion
we have 1<ι < 00. If condition c) is satisfied, let q = q\. Note that from
(8.3.7) we have h < 00. Since β{χ)1{β{χ))/xq2 [ implies xl(x)/a(x)q2 |,
by (8.3.10) we have
y> l(j) < n1+gV*Z(n) ~ .-1-21Z21 = Q/ ^H \
^ OLqi (j) ~ α«1 (η) ^ J 'а91(п)/'
j=n x" ' v ' j=n

8.3 Complete convergence for (^-mixing sequences 207
It follows that I4 < 00. Therefore I2 < 00. The proof of Theorem 8.3.2 is
completed.
First we show that (8.3.8) implies (8.3.9). Put dn = Y^rT1 l(j). We
Proof of Theorem 8.3.3
I
have
.*(n)-*([n/2])nf _„_/?(,%)
η
oo 2^+1-l
g'w-'(i^i>P{su,,g№>>a
n=2 n ^i>n г
<£ Е-!ВДИр{вир^)>л
j = l n=2J
00 2^+1-l 2^-1
/?(S)
OO OO
< £ 2-^Ц· - 2^-χ) Σ Р{2Ь m«+1 /3(5.) > ε2*}
oo
<Σ2~4ρ{ ш« /3(*)>ε2*}
. , ^2к<г<2к+г
fc=l —
00
<Σ4 Σ ^Р{?аДО,)>£2'}
jfc=l 2*+1<η<2*+2-1 -
<4y lMpfmaxp(Si) > ^) < 00. (8.3.19)
^—"ί П { г<п 2 J
n=l ~
This proves that (8.3.9) holds true.
Now we show that (8.3.2) holds true. From (8.3.19) it follows that
00
Σ Z(2fc)P{max \Xi\ > a(e2k)} < oo (8.3.20)
for any ε > 0, which implies
Pimax \Xi\ > a(e2*)| —► 0, as к -> oo. (8.3.21)
Since φ{ρ) —> 0 as ρ —> oo, there exist A?o and po such that for any к > ко
φ(ρ0) + PJmax |X;| > a(e2k)\ < 1/2.
^ i<2k '
Then, as in (3.2) of Lai (1977) we have
P{ max \Xi\ > a{e2k)\ > —P{\XX\ > a{e2k)}.
^l<i<2k > 2po

208
Chapter 8 Laws of Large Numbers and Comolete Convergence
Whence
oo
^/(2fc)2fcP{|X1|>a(s2fc)}
k=l
oo
< c^2 l(2k)P{ maxfc \X{\ > c*(e2fc)} < oo. (8.3.22)
/e=l
On the other hand
Εβ(Χ1)1(β(Χ1))
OO
< /(1) + Σ 2jl(2j)P{2j~1 < β(Χχ) < 2j}
OO
< /(1) + Σ 2ji(2J)P{|Xi| > a(2^-1)}.
From (8.3.22) we obtain Εβ{Χ1)1{β{Χι)) < oo. (8.3.2) is proved. This
completes the proof of Theorem 8.3.3.
8.4 Complete convergence for p-mixing sequences
The conclusions of complete convergence for a /9-mixing sequence have
not arrived completely as for the (^-mixing case, but some ideal sufficient
conditions can be given.
Definition 8.4.1. A function l(x) > 0(x > 0) is said to be quasi-
monotone non-decreasing, if
limsup sup l(t)/l(x) < oo.
x—юо 0<*<ж
A function l{x) is said to be quasi-monotone non-increasing, if
limsup sup l(t)/l(x) < oo.
X—ЮО t>X
Throughout this section, we assume that l{x) is a positive, even and
quasi-monotone non-decreasing function, β(χ) is a positive even function
and β(χ)/χθ, χ2 Jβ{χ) are monotone non-decreasing for some θ > 0. Denote
a(x) = Ίηνβ(χ).
Shao (1989c) proved the following theorems.

8.4 Complete convergence for p-mixing sequences
209
Theorem 8.4.1. Suppose that β(χ)1(β(χ)) and χ2~ε° / β(χ)1(β(χ)) for
some 0 < εο < 1 are quasi-monotone поп-decreasing functions, {Xnin >
1} is a p-mixing sequence with a common distribution and EX\ = 0,
Εβ(Χ1)1(β(Χ1)) < oo. //
Ε р(2П) < °°. (8-4Л)
n=l
then for any ε > 0
l{n)
У -^-P{ max fi(Si) > en) < oo. (8.4.2)
Theorem 8.4.2. Suppose that there exist q > 2, go > 2 and 5 > 0 зг/с/г,
that χς/β(χ)1(β(χ)), β(χ)1(β(χ))/χ2 are quasi-monotone поп-decreasing
functions, and l{x) > χδ(χ > 0),
Σ°° l(n) /η1/2 \яо , A .
-^(-7-7) <oc· (8·4·3)
n=l ч '
Let {Xn} be a p-mixing sequence with a common distribution and EX\ = 0,
Εβ(Χ1)1(β(Χ1)) <οο. If
OO
Σ P2/r(2n) < 00 (8.4.4)
n=l
for some r > q, then (8.4-2) holds.
Corollary 8.4.1. Let {Xn,n > 1} be a p-mixing sequence with a
common distribution and EX\ = 0, .ElXil^/idXil1/01) < 00 for ρ > 1, pa >
1, a > 1/2 and a slowly varying function h{x) > 0. //
00
]Гр2/г(п)<оо (8.4.5)
n=l
for r = 2 Wien 1 < ρ < 2; r > p, when ρ > 2, йеп
n=l
/or any ε > 0.
00
V npa-2u(n)P{ max |S;| > εηα) < oo (8.4.6)

210
Chapter 8 Laws of Large Numbers and Comolete Convergence
An immediate consequence of the above complete convergence result
is the following Marcinkiewicz-Zygmund law of large numbers.
Corollary 8.4.2. Let {Xn,n > 1} be a p-mixing sequence with a
common distribution and EX\ — 0, £"|Xi|p < oo, 1 < ρ < 2. Assume that
oo
Σ ο(2") < °°·
n=l
Then
lim Sn/n1/p = 0. a.s.
п—юо
Corollary 8.4.3. Let {Xn,n > 1} be a p-mixing sequence with a
common distribution and EX\ = 0, i£|Xi|p/i(|Xi|p) < oo, 1 < ρ < 2, and h{x)
be a monotone поп-decreasing slowly varying function. Suppose that
oo
£>(2")<oo.
n=l
Then for any ε > 0
у Mp( max |S.| > εηι/ρ\ < <».
*-*ί η li<t<n ' "" J
n=l
If Е\Хг\р+6 < oo for some δ > 0 instead of the condition Е\Хг\рк
dXil1/6*) < oo in Corollary 8.4.1, condition (8.4.5) can be deleted.
Furthermore Kong and Zhang (1994) pointed out that (8.4.6) also holds true
for non-identically distributed random variables.
The proofs of Theorems 8.4.1 and 8.4.2 need the following lemma.
Lemma 8.4.1. Let {ξη,η > 1} be a p-mixing sequence with Εξη = 0,
Eg(\£n\) < oo? where g(x) is a function for which there exists a constant
0 < С < oo such that sup^^ -π < С'-т-у for any t > 0. Denote Tk(i) =
Ylj=k+i£j' Then for any #1,(72 > 2 there exists a constant К depending

8.4 Complete convergence for p-mixing sequences 211
only on qi,q2 and p(-) such that
р(тах|Т0(г)| > x\
{ i<n >
η
< Σ рШ ^ A) + K{x~qi {пЯ1/2
i=l
[log n]
■ exp(K Σ р(Г))
г=1
•mzx№im<B)\\Y
г<п
[log n]
+ nexp(tf Σ P2/qi(^))
г=1
•log*(2n)max£|fc|*J(|fc|<B))
г<п /
[log η]
+ х-*2(п*2/2ехр(д: ]Г ,9(2*))
■тах\\Ь1(В<\Ы<А)№
г<п
/о Р°6 П]
+(lY2/2keW(Kj:Py-w)
г=1
•тахВД92/(Ж|£|<Л))
г<п /
[log η]
+ x-2-np2(fc).exp(ii Σ Р&))
г=1
• log4 g] · max £#/(S < |fc| < Л)}
/or απ?/ к (< η), χ > 0 and Л > 5 > 0 which satisfy the following
conditions
^ max £<K|6|)/(|&| > A) < x, (8.4.7)
^^ max £<,(|&|)/(|&| > Б) < χ. (8.4.8)
Proof. For simplicity, we assume that {£,&,г > 1} have a common

212
Chapter 8 Laws of Large Numbers and Comolete Convergence
distribution. Put
61 = 6/(161 <£)--Efc/(l6l<£);
62 = iJ{B < 161 <A)- Ε&Ι{Β < 161 < A);
Ьз = ШЫ>А)-ЕЬ1№\>А);
i
Tik = Y^£jk, k = 1,2,3.
It is clear that
р{тах|Т0(г)| > x\
y- i<n J
<p\max\Ta\ > f } + P{max|Ti2| > 7}
l. i<n 4 > <- г<п 4 J
+ р{тах|Гй|>£}
^ г<п Ζ ■>
=: /ι + /2 + /3. (8.4.9)
From (8.4.7)
/8<Ρ{ΣΙ&№>Λ)>|
-f>|6|/(|6l>A)}
г=1
< ^{ΣΙ6ΙΑΙ6Ι >^)> §
г=1
-Σ^μι6ΙΚ(Ι6·ι>λ)}
<^{ΣΙ6|/(Ι6Ι>^)>|}
г=1
η
<Σ^(Ι6|>^)· (8.4.10)
г=1
By Lemma 2.2.5 and Lemma 4.1.2, there exists a constant K\ depending

8.4 Complete convergence for p-mixing sequences
213
only on qi and p(·) such that
[log n]
Ь^Кгх-ъ^МткВШехр^Кг £ p(2<))
i=0
+ nlog*(2n)|K/(K|<B)||«
[log n]
i=0
(8.4.11)
In order to estimate /2, put
where
Denote
Then
(2t+l)fc
Yi= Σ &2, ί = 0,1,···,ρι,
j=2ik+l
2(i+l)fc
Zi= £ fc2, < = 0,l,-.-,p2 (8.4.12)
j=(2t+l)fc+l
Λ = [(=-!)/»].«= [(=-4/2].
h<P{ max |Wi|>^)+p{ max |И?| > ^-}
~ ^0<i<pi ' ' — 12 J l0<t<p2 12 J
7
{Ι ι 7* л
max max Y^ £/o > — \
o<i<[n/k] ik+i<j<u+i)k \ ,r? л I 12 J
:[n/fc] ik+i<j<(i+i)k I tj7^+l
=:h + h + h-
(8.4.13)

214
Chapter 8 Laws of Large Numbers and Comolete Convergence
By Lemma 2.2.5 and (8.4.8) we have
(t+l)fc
J—lK-\-L
-Е\Ь\1(В<\Ы<А)
(t+l)fc
>^-2 Σ E\b\I(B<fa\<A)}
j=ik+l
(t+l)fc
S2-G]o<^4p{.?lf,l^<ifel<^)
L ' J з=гк+1
-Е)Ь\1(В<\Ь\<А)>£\
K2[l]x-q2(k^2\№B<\t\<A)\\f
[log?
•exp(tf2 Σ P(2J'))
t=0
[log n]
+ ЩЦ{В < \ξ\ < A)\\%exV(K2 Σ P2/92(2J')))
г=0
< K2x-q2(nq2/2Ш(В < \ζ\ < A)\\f
[log n]
■exp(K2 "£ P(2J))
+п-мцв<м<а)\\*
[log n]
•exp(tf2 Σ Р2/Я2(У))· (8.4.14)
i=0
Now we estimate /4. Denote T{ — a(£j,j < 2(г + l)fc), г = 0,1, · · · ,ρι,
.F_i to be a trivial σ-field. Put
г
ui = Yi - ВД-Fi-i), Οί = Σ Uj,
3=0
i
Hi = Y/E(Y3\Jr3-i), г = 0,1,···, px.
3=1
It is easy to see that
h < P{ max \GA > £-\ + P{ max |Я;| > — 1
=: /7 + /8. (8.4.15)

8.4 Complete convergence for p-mixing sequences
215
Note that \Ui,Ti,i > 0} is a martingale difference sequence. Hence by
the maximum velue inequality for a martingale difference sequence, the
Marcinkiewicz-Zygmund inequality and Lemma 2.2.5, we have
ы&ГШ'*™
141
-Q2
<K3x-q2-pf/2 max Е\ЩЯ2
0<i<pi
< K3x-<» .p*'2(k*'2MI(B < |e| < A)\\
[log n]
■exp(K3 Σ ρ(2ή)
i=0
+ k\№B<\t\<A)\\*
[log n]
•exp(tf3 Σ^(2')))
i=0
<Κ3χ-<»(η<»/2\\ξΙ(Β<\ξ\<Α)\\?
[log n]
•exp(tf3 Σ p(2*))
г=0
+ Qq2/2k.E\^4(B<\C\<A)
[log n]
ехр(к3 Σ P2/92(2J'))· (8.4.16)
г=0
By imitating the proof of Lemma 2.2.2 and using Lemma 2.2.5, there exists
a constant К\ depending only on p(·) such that
г+m 2
E{ Σ ВД^-ι))
i=t+i
< K4mkp2(k)E£2I(B < \ξ\ < A)
[log n]
•log2(2m)exp(tf4 Σ Р&))· (8'4Л7)
i=0
From Lemma 4.1.2 it follows that
Ε max Η2 <ZKiPlkp2{k)\ogi{2pl)EiiI{B < \ξ\ < A)
l<i<pi
[log n]
•exp(tf4 Σ Ρ&))- (8·4·18)
г=0

216
Chapter 8 Laws of Large Numbers and Comolete Convergence
Thus we obtain
/8 < ^4x-W(*)log4[^]^2/(S < \ξ\ < A)
[log n]
■ехр(к4 Σ p(2*)). (8.4.19)
г=0
(When [n/k]. < 2, we have pi = 0 and hence Ig = 0, i.e., (8.4.19) holds
true; when [n/k] > 2, (8.4.18) implies (8.4.19).) Prom (8.4.15), (8.4.16)
and (8.4.19) it follows that
h < K5x-^(n^2UiI(B < |fc| < A)\\f
[log?
exp(K5 Σ Р&))
г=0
+ {1)Ч2'2к\\тв<ы<А)\\1
[log η]
•exp(K5 £ р2/чЧ?))
+ K5X-2np2(fc)log4[^]||^7(B < |б| < A)\\l
■ иа{в<и<А)\\1
[log n]
.exp(tf5 Σ A*2'))· (8-4.20)
By the same way, we have also (8.4.20) for /5. Lemma 8.4.1 is proved.
Proof of Theorem 8.4.1.
From the assumption that 1(χ),β(χ)1(β(χ))/χ,χ2~ε°/ β(χ)1(β(χ)) are
quasi-monotone non-decreasing, it follows that there exists a constant С >
0 such that for any t > χ > 0
l(x) < Cl(t),
β(χ)1(β(χ))/χ < C/J(t)i(/J(t))/t,
χ2-ε°/β(χ)1(β(χ)) < α2-ε°/β(ί)1(β(ί)).
By (8.4.22), (8.4.23) and the monotonicity of β(χ) we have
a(x) 1 a(t)
xl{x) ~ CtUjt)'
a2-£°(x) a2-£0(t)
xl(x) ~ tl(t)
(8,
(8,
(8,
(8,
(8,
4.21)
4.22)
4.23)
4.24)
4.25)

8.4 Complete convergence for p-mixing sequences
217
Let Cn = log-8/£° η and take A = a(n),B = a(nCn),k = [nCn],x =
a(en),g(x) = β(χ)1(β(χ)) in Lemma 8.4.1. Then (8.4.7) and (8.4.8) are
satisfied for large n. In fact, for η > Ι/ε, from (8.4.24) and (8.4.21) we
have
^^-Εβ(Χ1)1(β(Χ1))Ι(\Χ1\ > a(n))
nlyn)
< ^^ Εβ(Χ1)1(β(Χ1))Ι(\Χ1\ > a(n))
εί[εη)
- 4С^П)Εβ(Χι)Κβ(Χι))Ι(\Χι\ > α(η))·
Noting that limn^oo Εβ(Χι)1(β(Χι))Ι(\Χι\ > a(n)) = 0, for large η we
obtain
4°Μη) > < a{£n)
ni{n)
This proves that (8.4.7) is satisfied. Similarly we can verify (8.4.8). By
Lemma 8.4.1, letting q\ = q<i = 2, we have
= V* iiulpf SjI > α(ε„)}
n=l
oo
<X)i(n)P{|A-!|>a(n)}
n=l
oo
n=l
oo
+ CE 4π£Χι2/(Ι^Ι ^ oc{nCn))\og2n
+ <t^jMiA№ls-(-))
n=l ч '
+ c Σ -li^nCnjfloglogn^lf/dl!! < a(n))
71=1 Ч '
=: Л + J2 + J3 + Ji- (8.4.26)

218 Chapter 8 Laws of Large Numbers and Comolete Convergence
From (8.4.21)
OO
n—1 j=n
oo j
j=ln=l
oo
<c^J7(j)P{j</3(X1)<j + l}
< οΕβ(Χ1)1(β(Χ1)) < oo. (8.4.27)
From (8.4.23), (8.4.25) and the monotonicity of χ/αθ(χ),α2(χ)/χ we have
~ l{n) log2 na\nCn)
32 ~ ° Σ «2(en)nCri/(nCn) Εβ(Χχ)1<β{Χχ))
n=l ^ '
oo 1
<c^-—^-<oo. (8.4.28)
Since Σ^χ p(2n) < oo we have p(n) < clog-1 n. Hence
J4 < cf ^P-Aog-^nEXll^ < a(n))
n=l v '
oo ι
^Σ , 3/2 <°°· (8·4·29)
At last we estimate J3. From (8.4.25) and x/a2(x) j, we have
f* /(j) = f> TO i£o/2 1
~* a2(j) ^ a2-£o(j) a£°(j) ji+eo/2
j
d(n)wW» ^ W2
- a2(n) ^
с n/(n)
~ ε0 α2(η)'

8.4 Complete convergence for p-mixing sequences
219
Therefore
~ αζ(η)
n=l ч '
oo ?/ \ η
+ Σ$πΣΕχϊ'ϋ<β{χι)<ί + ι)
n=l ^ ' j=l
OO OO ι/ \
.7 = 1 n—i ^ '
j=Ln=j
oo
■ ^ CE ЩехЩ* < β(Χι) < J + 1)
£ί α Ι?)
< οΕβ(Χχ)1(β(Χι)) < oo. (8.4.30)
Theorem 8.4.1 follows from (8.4.27)-(8.4.30).
Proof of Theorem 8.4.2.
From the assumption that l(x), xq / β(χ)1(β(χ)) and β(χ)1(β(χ))/x2 are
quasi-monotone non-decreasing it follows that there exists а С > 0 such
that (8.4.21) is satisfied and for any t > χ > 0 we have
cfl{x) _α<(ί) β(χ)1(β(χ)) ГЩ0) ,R , _
1вд-с"й(0' ^ -c—«δ—· (8·4·31)
Let Cn = η ei, where e\ = 2(i+6)> ^ @ ~ a(nCn),k = n,g(x) =
β(χ)1(β(χ)). Take χ = α(εη) in Lemma 8.4.1. Then (8.4.7) and (8.4.8) are
satisfied. In fact, by l{x) > χδ we have for large η
48Ca(»a)
CJ(nCn)
<
48Ca(en)
<- %$$w*m*))
< "ns/2 Έβ{Χχ)1{β{Χι)).
Thus (8.4.8) is satisfied for large n. By the same way (8.4.7) is satisfied as
well.

220
Chapter 8 Laws of Large Numbers and Comolete Convergence
By Lemma 8.4.1 with q\ = qo + q + 4, <?2 = г we have
l(n)
У ^Ытах/?($)>еп}
^f-\ Π ^ г<П J
oo
<c£i(n)P{|Ai|>a(n)}
n=l
n=l
oo // \ [bgn]
• log91 η£|ΛΊ|*/(|Λ:ι| < a(t»C„)))
+ сЕ^К(^?/(|Хг|>а(пСп))Г/2
n=l ^ '
+ nE|X1|r7(|X1|<«(n)))
=: Li + L2 + L3. (8.4.32)
As in the estimation of J\ in the proof of Theorem 8.4.1, we have
Li < oo. (8.4.33)
Noting that ехр(лГЕ!=ёГ1Р2/<г1(20)
log91 η is a slowly varying function,
£*' T, and using (8.4.31) and condition (8.4.3), we have
L2 < с Σ -^T-yi/2E\Xi\qI(\Xi\ < oc{nCn))
^—ί aqi(n)
n=l ч '
^ ^ l{n)n^2a^{nCn)
<tv,./»°"W
- z-j a^-9(n)
n=l ч у
oo
n=l
oo
<οΣη~1-ει <°°· (8.4.34)
n=\

8.4 Complete convergence for p-mixing sequences
221
Next we estimate L3. From (8.4.31) it follows that
~ l(n) ,nEXll{\Xx\ > a(nCn))y/2
hin ^ <*» '
l(n) ar(nCn)
£ϊ η CTn/2l^(nCn)ar(n)
^ ^ l(n)n ar(nCn) 1
< c У —;
^ of {η) nCJ(nCn) nCrnl2-llr/^{nCn)
00 ,
<CY i
- tinCrnl2-\nCn)W-W
OO
< с Σ η-ι~{ΐ^τ)δ < 00. (8.4.35)
n=l
By (8.4.35) and (8.4.31) we obtain
l(n)
^—^ ar(n)
71=1 Ч '
" tx α"(η)
1{η)
n=l ^ ' 7 = 1
J
n"
^-^ al{n) η 2 OLr(l)
<CL·
'^Ί <*q(n) ar~q(n) n1+(r-9)/2
~ ~ nl(n) n(r-«)/2 1
4-i ^ a*(ra) ar~9(n) ni+(T-g)/2
j=ln=j ч ' x '
■E\X1\4(j<p(X1)<j + l)
< βΕβ(Χι)1(β(Χι)) < 00. (8.4.36)
Then Theorem 8.4.2 follows from (8.4.32)-(8.4.36).
In order to prove Corollaries 8.4.1 and 8.4.2, we need the following
lemma.
Lemma 8.4.2. Let h(x) be a positive slowly varying function. Then
xeh(x) is a quasi-monotone поп-decreasing function for any ε > 0.

222
Chapter 8 Laws of Large Numbers and Comolete Convergence
Proof. By the property of a slowly varying function, we have
Г KX) у · r KX) Л
lim sup , /^лг^тч = lim inf f , .,^14 = 1.
Therefore there exists an TV such that for m > N we have
sup fe(g) <2£, 2-£< inf гт^тг. (8.4.37)
2™<Χ<2-+1 /l(2™+X) - - 2m<x<2m+l h(2m + 1) K '
For any t > χ > 2N, let MUM2> N such that
2Ml < χ < 2Ml+1, 2M2 < t < 2M2+1. (8.4.38)
From (8.4.37) and (8.4.38) we have
хЩх) < χ* sup _|g> Λ(2*+ΐ)
< ^2e(M2-Ml+1)/i(2M2+1)
< x£2£(M2~Ml+Vh(t)
= 2ε ■ t£h(t).
This proves that xeh(x) is a quasi-monotone nondecreasing function.
Proof of Corollary 8.4.1. From pa > 1 and Lemma 8.4.2 it
follows that npa~1h(n) is quasi-monotone nondecreasing. If 1 < ρ < 2,
xph{x), x2~p/h(x) are quasi-monotone nondecreasing. Then Corollary 8.4.1
follows from Theorem 8.4.1. If ρ > 2, xp~2h{x) and xr~p/h(x) are quasi-
monotone non-decreasing. Then Corollary 8.4.1 follows from Theorem
8.4.2. If ρ = 2, lim^oo η~εΕΧ\ -I(\Xi\ < na) = 0 for any ε > 0, and
hence Corollary 8.4.1 can be obtained by repeating the proof of Theorem
8.4.2.
Proof of Corollary 8.4.2. Obviously xp~1h(x) is increasing. By
Lemma 8.4.2, x2~p/h(x) is quasi-monotone non-decreasing. Hence
Corollary 8.4.2 follows from Theorem 8.4.1.
8.5 Complete convergence for α-mixing sequences
For the complete convergence of α-mixing sequences, Hipp (1979)
obtained the following result:

8.5 Complete convergence for α-mixing sequences
223
Theorem 8.5.0. Let
-<a<l,2<r<oo, 1/a < ρ < r, {Xn, η > 1}
be a strictly stationary α-mixing sequence of random variables with EX\ =
0,E\Xi\r < oo. Assume that
oo
У аг'в(п) < oo for some θ > fo + —^1 · -J^—, (8.5.1)
~y L r-pJ pa-1
Then
oo
V na-2P( max |5<| > εηα) < oo for any ε > 0. (8.5.2)
n=l
However, an example contrary to Hipp's conclusion was given by Berbee
(1987) when r = oo, i.e., in the case of X\ bounded.
Shao (1993c) proved the following theorem.
Theorem 8.5.1. Let 1/2 < a < 1, 1/a < ρ < r < oo, {Xn,n > 1} be
an α-mixing sequence of random variables with EXn = 0, supn>1 ||Xn||r <
oo. Assume that
a(n) = 0(n-rip-1)/{r-p) log_/3 n) for some β > rp/(r - p). (8.5.3)
Then (8.5.2) holds true.
An immediate consequence of Theorem 8.5.1 with ρ = a = 1 is:
Corollary 8.5.1. Let 1 < r < oo, {Xn,n > 1} be an α-mixing
sequence of random variables with EXn = 0,supn>1 ||Xn||r < °o· Assume
that
a(n) = 0(\og~P n) for some β > r/(r — 1).
Then
OO -j
У^ —P\ max \Si\ > εη\ < oo for any ε > 0.
n=l
Particularly, we have
Sn/n —> 0 a.s.
The proof of Theorem 8.5.1 will needs the following lemmas.

224
Chapter 8 Laws of Large Numbers and Comolete Convergence
Lemma 8.5.1. Let {Xn,n > 1} be a sequence of random variables
with EXn = 0 for every η > 1. Then
PJmax ISA >x\ < Ax~l Υ Е\Хг\1(\Х{\ > с)
г=1
+ 4ха + 323псх~1а(к) (8.5.4)
for any α > 1, χ > 1, с > 0 and integer к satisfying
l<k<x/(64ac\ogx) (8.5.5)
and for some s > 2,
(Σ HXJdXil < c)||f) E«1_2/S« < x2/(323aloga;). (8.5.6)
t=l г=0
Proof. Let
Xi = XiI(\Xi\ < c) - EXiI(\Xi\ < c),
(2г+1)&Лп
УМ= £ X,-, i = 0,l,...,gi:=g--],
j=l+2ifc
2(г+1)&Лп
Уг,2 = Σ Xj9 ι = 0,1,.··,^:=[^-ΐ],
j=l+(2t+l)fc
г г г
j=l j=0 j=0
It is easy to see that
max |Si| < max \S{\
1<г<п 1<г<п
η η
+ Σ\Χί\Ι{\Χί\> с) + YjE\Xi\I{\Xi\> с)
г=1 г=1
and
Р{ max \Si\ > χ) < P{ max 15.1 > x/2)
η
+ Ax~x Σ E\xiW*i\ > c)· (8·5·7)

8.5 Complete convergence for α-mixing sequences
225
max
1<г<г
Since
jc [SA < max \Tii\-\- max |Ti2| + 2fcc
Cn ' ' ~ 0<i<qi ' ' ' 0<i<92 '
< max |Tvi|+ max IT; ok+ #/32
0<г<91 ' 0<г<92
by (8.5.5), we have
Ρ\ max ISA > χ/2}
11<г<п >
<p(max \Тц\ > χ/δ) + p{ max \Ti2\> x/%\
=: J1+J2. (8.5.8)
We first estimate I\. The estimation of I2 is completely similar. Put
(?_! = σ(Ω), Gi = σ(Χά, 1 < j < (2г + l)fc),
г
^г = Ή,ι -£(^,ι|<3;-ι), Ε/, = ]Ргу, г = 0,1,···.
i=o
Then
/^pjf^l^y^l^oi^x/ie}
г=0
+ p( max |ui| > х/1б)
10<г<?1 J
=: /3 + h (8.5.9)
and {t/i, Gi, г > 0} is a martingale with \щ\ < 2kc for every г > 0. Noting
that for each real t and г > 1,
E(e'»'|Gi_,) = l + f;£;(fi5)i|Gi_1)
< exp(i2e2l<lfcc£;(u?|Gi_1))
<exp(i2e2ltlfcc£;(yi21|Gi_1)),
we find that {exp(tUi - t2e2^kcZUoE(Yli\Gi-i))^i,i > θ} is a non-
negative supermartingale for every t and hence
i 1
P{ max exp(«7i - t2e2^kc Σ E(Yj\i\Gj-i)) > у} < ~ (8.5.10)

226
Chapter 8 Laws of Large Numbers and Comolete Convergence
for у > 0, by the maximum inequality of the non-negative supermartingale.
Take t = (32alogx)/x in (8.5.10). By (8.5.10) and (8.5.5), we have
P{ max Ui > х/1б)
l 0<i<qi J
< Fornax ехр(од - t2e2WkcJ2E(Yli\Gi-i))
j=0
+ P{ max exp(tUi - t2e2^kc £ Я(У* ^--х))
-г-91 ,·=ο
,xt t2e2Wkcx2 u
-eXPVl6 4(32)2alogJ/
91
si ^е21^сж2
16 + 4(32)2alog:
/ χτ τ e ' ' χ \
+ eXH~l6 + 4i32)2alogJ
91 χ2
< P{g^«ilGi-x) > 5(^fe}+ *""· (8-5Л1)
Using Lemma 1.2.4, one can obtain
qi к qi (2j+l)k
ΈΕΥΙι<ΚΈ^~2/^))Έ Σ \\Xii(\Xi\<c)\\2s
j=0 г=0 j=0i=2jk+l
<А§Га'-21°(г))^\\Ха{\Х>\<с)
<
2
s
г=0 г=0
2
8(32)2alogx
by (8.5.6). Hence
Р{;|Е№-.)>щ^}
<8(32)!;'osxi:^^.i^-.) - иг.1· ««·"«
j=0

8.5 Complete convergence for α-mixing sequences 227
Write £, = EiYf^Gj-x) - EY?V We find that
Efa\ = E{Y}tX - EYfoBfstfi < 4(kc)2a(k) (8.5.13)
by Lemma 1.2.4 again. Inserting (8.5.13) into (8.5.12), we obtain
^ (32)3anc2ka(k) log χ (S2)2nca(k)
< < (8.5.14)
χ2 χ
by (8.5.5). Now a combination of (8.5.11) and (8.5.14) yields
P{ max Щ > 4:) < x~a + (32)2ncx_1a(fc).
^0<i<qi 16 J
Similarly, we have
Pi max Щ < - — } < x~a + (32)2ncx_1a(fc).
^0<i<qi 16 J
Hence
h < 2x~a + 2(32)2ncx_1a(fc). (8.5.15)
Also, one can get that
E\E(YiA\Gi-i)\ = EYiABgnWYidGi-i)) < 4kca(k)
and hence
h < Uncx~la{k). (8.5.16)
It follows from (8.5.9), (8.5.15) and (8.5.16) that
h < 2x~a + 3(32)2ncx~1a(k). (8.5.17)
Similarly, we have
h < 2x~a + 3{32)2 ncx~la{k). (8.5.18)
This proves (8.5.4) by (8.5.7), (8.5.8), (8.5.17) and (8.5.18).
Lemma 8.5.2. Let {Xn,n > 1} be an α-mixing sequence with
EXi = 0, \\Xi\\v<D and a(i) < C0i~Tlog~Aг

228
Chapter 8 Laws of Large Numbers and Comolete Convergence
for г > 1 and for some 1 < ν < oo, Co > 1, r > 0 and rea/ λ. Then there
exists a finite positive constant К depending only on ν, τ, λ and Co such
that
Pi max \Si\ >x)< Kn{D/x)^r^l^Thog^-l^r-x^^r\xlD)
(8.5.19)
/or any χ > KDn1'2 log1+lAl/2n.
Proof. We assume , without loss of generality, that D = 1. It
suffices to show that there exists a constant К such that
ΡI max ISjl > χ >
ll<i<n J
< Χηχ~Ι/(τ+1)/(Ι/+τ) log(,/-1)(T-A)/(,/+T) χ (8.5.20)
for any χ > Kn1/2log1+W2n. Take
С = 2xT^+Thog(x-TV^+T) x,
a = τ + 2, fc = [x/(64aclogx)]
in Lemma 8.5.1. Assume that
ϋΓηχ-Ι/(τ+1)/(Ι/+τ) log^-1^"^/^^) χ < 1. (8.5.21)
Otherwise, (8.5.20) is trivial. If the conditions of Lemma 8.5.1 are satisfied,
then (8.5.20) will follow from (8.5.4) immediately. So we need only to
verify (8.5.6). In what follows we denote by K\ the finite positive constant
depending only on j/, τ, λ and Co, whose value may be different from line
to line. If 1 < ν < 2, then
i=l i=0
< 2nkc2~v < xncl~v/{Z2a\ogx)
■(32)221-unx-^T+1^t/+T^
x2
(32)3alogx
. iog("-i)(T-A)/(v+T).
x2
< , ., , (8.5.22)
~ (32)3alogx v ;

8.5 Complete convergence for α-mixing sequences
229
by (8.5.21). When и > 2, we have
i=l i=0
< n(l + Co Σ t-T(1-2/"> log-^1"2/") i)
i=X
< Kin^og1+im-2/u) к + k-T^-2^+1 log-^1"2^ k)
< Kin(logl+^ χ + {x/{c\ogx))~<x~2l^+l log-A(1-2/l/) x)
< 7^fr—{K.nx-4oe+^-^ χ
(o2)oa\ogx
+ Kxnx-^+W"^ log^"1)^-^/^^) x)
< τ—τί (8.5.23)
~ (32)3alogx k У
by (8.5.21) and χ > Kn1/2 log1+|A|/2 n. Now we conclude from (8.5.22) and
(8.5.23) that (8.5.6) is satisfied. This completes the proof of the lemma.
Proof of Theorem 8.5.1. By Lemma 8.5.2, we have that for any
ε > 0, there exists a positive constant К such that
P{ max ISA >εηα)
l 1<г<п /
< urri1~Q:r(1+r(P~1)/(r~P))/(r+r(P~1)/(r~P))
. log(l-r)(i9-r(p-l)/(r-p))/(r+r(p-l)/(r-p)) n
= if η1-ρα log"1"(r"p)^"rp/(r"p^/r η
which yields (8.5.5) immediately by (8.5.6), as desired.
The example below shows that Theorem 8.5.1 does not remain valid,
if the assumption β > rp/(r — p) is replaced by β > r/(r — p).
Example 8.5.1. Let 1 < ρ < r, 1/p < a < 1. Put
a(r-p) fl(p-l) ■ -1
a = — , ο = , a —
r(l — a) + pa — l' r — p ' r — 1 — a(r — p)'
З(ж) = xalogdx, ж > 0,
η
G(0) = 0, G(n) = J>(0], η = 1,2,···,
/(ж) = (5(a;))'-(P-1)/('-^(log5(x))r/('-P)loglog5(a;), χ > 0.

230
Chapter 8 Laws of Large Numbers and Comolete Convergence
Let {Yn,n > 1} be an independent sequence of random variables with
Define a sequence {Xn, η > 1} by Xn = Yj for G(j — 1) < η < G(j). Then
{Xn,n > 1} has the following properties.
EXn = 0, E\Xn\T = 1, (8.5.24)
a(n) = 0(n~rip~1)/{T-p) log-p/(p-p) n(loglogn)-1), (8.5.25)
OO
Σ npa-2P{\Sn\ > εηα} = oo for any ε > 0. (8.5.26)
n=l
Example 8.5.2. Let r > 1. Put
η
a = (r-l)/r, »(n) = [na-1exp(na)], G(n) = X)^i).
г=1
Let {Υη,η > 1} be an independent sequence of random variables with
P{yn = ±n1/4og1/'-n} = ——, Р{УП = 0} = 1- Х
2n log η' η log η
Define Xn = Yj for G(j - 1) < η < G(j). Then {Xn,n > 1} has the
following properties.
EXn = 0, ΕΊΧηΙ7, = 1,
a(n) = Oilog-^^-^Ooglogn)-1),
oo 1
У^ —P{|5n| > en} = oo, for any ε > 0.
n=l
Remark 8.5.1. We now can discuss whether Theorem 8.5.0 is true
or not in the case of r < oo. Assume that 1 < ρ < r/2. Then (8.5.1) will be
satisfied with θ > 8 provided ρ large enough. But Example 8.5.1 says that
the mixing rate n~~(p-1"2 is required at least. This means that Theorem
8.5.0 is quite possibly not true. Unfortunately, {Xn,n > 1} in Example
8.5.1 is not strictly stationary. Shao (1993c) conjectured that there is
a strictly stationary α-mixing sequence satisfying (8.5.24), (8.5.26) and
a(n) = 0(n~r(p~1)/(r-p) log-1 n). He also conjectured that the assumption
β > rp/(r — p) in Theorem 8.5.1 can be replaced by β > r/{r — p).

8.6 A further discussion on the complete convergence
231
8.6 A further discussion on the complete convergence
for partial sums of a mixing sequence
Let {Xn,n > 1} be a sequence of i.i.d.r.v.'s, Sn = Y%=iX%. Assume
that positive functions H(t) and ψ(ί) are defined on (0, oo) and H(t) |
oo(t —► oo), ip(t) = $^ф(и)(1и, t > 0. Denote
oo
«/(e)=]T/((|Sn|>etf(n)),
n=l
V(e) = sxip{(\Sn\-eH(n))+/£},
n>l
χ(ε) = sup{n > 1 : \Sn\ > eH{n)},
where ε is an arbitrary positive number. Three problems posed by Pro-
horov are as follows:
(i) Is
OO
53 i>{n)P{\Sn\ > e#(n)} < oo, Ve >0 (8.6.1)
71=1
equivalent to
Εψ(ν{ε)) < oo, Ve > 0?
(ii) Is
OO
53 Ψ(η)Ρ{ max \Sk\ > εΗ(η)} < oo, Ve > 0, (8.6.2)
Г
equivalent to
Kk<n
n=l — —
Εψ(ΐηνΗ(η(ε))) < oo, Ve > 0?
(iii) Is
53 φ(η)Ρ{8ηρ \Sk\/H(k) > ε} < oo, Ve > 0, (8.6.3)
n=l k^n
equivalent to
Εφ{χ{ε)) < oo, Ve > 0?
Let
Mi = {φ : φ(χ) > 0,x € [1,оо),Э<5!,<52 > 0, such that
χι~6ιφ{χ) Τ, е-б2Хф(х) i, χ -»· oo}.

232
Chapter 8 Laws of Large Numbers and Comolete Convergence
Sirazgimov and Gafrov (1987) discussed these problems and showed
that under the following assumptions:
lim sup H(Cx)/H(x) < oo, VC> 1; (8.6.4)
X—ЮО
liminf P{Sn > -εΗ(η)\ > 0, Υε > 0 (8.6.5)
η—юо
and
φ £ Μχ or ^(ж), as χ —> oo,
the following three conclusions are equivalent
P(V>,#,e)
OO
:= J] V>(n)^{5n > eH(n)} < oo, Ve > 0, (8.6.6)
n=X
М(ф,Н,е)
OO
:= Σ V^P^max 5fc > еЯ(п)} < oo, Ve > 0, (8.6.7)
n=
Sty,H,e)
Kk<n
n=l — —
= Σ V>H^{sup Sk/H(k) > ε} < oo, Vs > 0. (8.6.8)
71=1 fc>
Su (1989) studied these problems and gave the following result.
Suppose that the following conditions are satisfied:
(A) There exist constants С > 0, to > 0 such that ψ(ί) < Οψ(ί) for
any t > t0;
(Β) ίφ[ί) | oo{t —> oo) and there exist С > Ο,^ο > 0 such that for
any t > to
[ иф(и)аи > C't2rl>(t)\
Jo
(C) 0 < /3i < H(t)/t < β2 < +oo or H(t)/t J oo(* -> oo) or #(*)/* 1
0 and H2(t)/t J oo(* -> oo);
(D) There exist a positive integer TV and δ2 > 0 such that for every
integer A: > 0
Σ ^(η)>(1+ &)]£>(")·
n=l n=l
Then (8.6.1) 4=^ (8.6.2) «=* (8.6.3) «=* £ty(*(e)) < oo.
Wang (1993) discussed Prohorov's three problems for a strictly
stationary /9-mixing sequence. Denote

8.6 A further discussion on the complete convergence
233
φ*(ΐ) = sup sup
к Ает^вет™+1,р(А)Р(В)>о
• шах{|Р(5|Л) - P(B)l \P{A\B) - P(A)\} (8.6.9)
p{x) = \{χφ{χ) - [|]^([f ]), x > 1, (8.6Л0)
Μί = {V> : ^0*0 > 0,ж > 1,ηψ(ή) -► oo,ra^(ra) > [ra/2]^([ra/2])}.
Wang (1993) proved the following theorem.
Theorem 8.6.1. Suppose {Xn,n > 1} is a strictly stationary p-mixing
sequence satisfying
liminf P{Sn > -εΗ(η)} > φ*(1) for any ε > 0. (8.6.11)
71—► OO
Suppose that ψ G Μ]*, Η (η) satisfies (8.6.4) and
00
53 V»(")p(") < oo (8.6.12)
n=l
and for ф € Mx* - Mi
pfe? ш -ε)
^cp{&lwk^-C£}< - = 1'2'-' (8·6·13)
<k>nH(k)
>Jinf
Я(*)
/or some С > 0. ΓΛβη /or any ε > 0 (8.6.6) <=> (8.6.7) «=*
Ξ(φ\Η,ε) <οο. (8.6.8х)
The proof of Theorem 8.6.1 will be given by several lemmas.
Lemma 8.6.1. If for any χ > 0
liminf inf P{Sn - Sk > -x\ > φ*(1), (8.6.110
n->oo k<n-l
then there exists a constant С > 0 such that for any у and η G N
p{maxSfc >y\ < CP{sn > у - x\. (8.6.14)
l* k<n > I J

234
Chapter 8 Laws of Large Numbers and Comolete Convergence
Proof. By the definition of φ*(1) and (8.6.IIх) we have
P{Sn >y-x)
> P\Sn >y- #,maxSfc > y\
у- к<п У
η
= У2 P\Sn >V~x,sk>y, max Si < y\
f—-' ^ г<к— 1 У
k=l
η
> У] P{Sn - Sk> -χ, Sk > y, max S{ < y\
f~-r l* г<к—1 У
k=l
η
> Y,(P{Sn -Sk> -x}P{sk > у max $ < y)
- ^*(l)PJ5fc > y, max Si < y\)
l* г<к— 1 J /
> CP\ma,xSk > у f, η = 1,2, · · ·.
I k<n У
Lemma 8.6.2. Let {ХП5^ > 1} be a strictly stationary sequence
satisfying (8.6.11) апапф(п) > С, п= 1,2,···. ΓΛβη (8.6.6) <=> (8.6.7).
Proof. We need only to prove (8.6.6) => (8.6.7). Take у = εΗ(ή),χ
= εΗ(η)/2 for any ε > 0 in (8.6.14). It is enough to prove (8.6.11'). First
we have
P{Sn > eH{n)} —► 0, as η -> oo. (8.6.15)
Otherwise there exist {щ},г and εο > 0, such that
P{Sni > е0Н(щ)} > r > 0.
Without loss of generality, we assume that щ+\ > Зп^, г = 1,2, · · ·. So by
(8.6.11) for any 2n; < η < Зщ, г = 1,2, · · · we have
P{S„ > ЦЩп)}
> P{sni > e0H(n),Sn - Sni > -jH(n)}
> P{Sni > e0H(n)}(p{sn - Sni > -Цн(п)} - φ*{\))
> P{Sni > e0H(n)}(p{sn-ni > -Цн{п - щ)} - ψ*{\))
>ri >0

8.6 A further discussion on the complete convergence 235
for some v\ > 0. It follows from ηψ(η) > С > 0, η = 1,2, · · · that
oo 3m
Ρ(ψ,Η,ε0/2)>Σ Σ Tp(rn)P{Sm>e0H(m)/2}
i=\ m=2ni
oo 3m
> ri ^2 X] тф(т)/т
i=l m=2ni
oo 3m *
>^<?Σ Σ 1 = οο>
г=1 77г=2тгг
which contradicts with (8.6.6). Therefore from (8.6.15) and (8.6.11) we
obtain that for any ε > 0
liminf inf P{Sn - Sk> -2εΗ(η)\
n->oo k<n-l
> liminf inf P{Sn > -eH(n),Sk < eHin)}
n—>oo k<n—l
> liminf inf (P{Sn> -eHin)}- P{Sk> eHin)})
n—>oo k<n—l
Lemma 8.6.2 is proved.
Lemma 8.6.3. Let {Xn} be a sequence of random variables, H(x)
satisfy (8.64) <*>nd пф(п) > [п/2]ф([п/2]), η = 1,2,···. Then (8.6.7) =»
(8.6.81).
Proof. Noting that ηψ(η) j and H(n)/H{2n) > δ we have
пф(п)-[п/2]ф([п/2})п( Sj
1SUP ui ■
<i>n H(i
пф{п)-[п/2Щ[п/2])< Si
*< η Ч>пЯ(г) - J
P{ SUP ttc\
3=1 n=2J %-Δ ν '
n=2
oo 2>+1-l

236 Chapter 8 Laws of Large Numbers and Comolete Convergence
oo Ί 2^+1-l
<Σ^Σ Η>(η) - [n/2M[n/2]))
3 = 1 n=2J
■TP\ sup -^r>e)
oo , 2'+1-1 , 2^-1
J=l n=2> n=2i~1
oo
.^PJsupS^stf^-1)}
oo 1 2fc+1-l
^Σ^ Σ ^(п)Р{8иР5,>бЯ(2^1)}
k=l Δ п=2к ^2к
oo 2fc+1-l
^2Σ Σ ^H^{sup5i>s(5^(2fc+1)}
fc=l n=2fc ^2к
oo
< 2 ^ ^(n)p{sup5i > εδ2Η(η)}.
n=2
i<n
By the arbitrariness of ε and (8.6.7) one gets Lemma 8.6.3.
Lemma 8.6.4. Under the conditions of Theorem 8.6.1, we have (8.6.8)
=Φ (8.6.7).
Proof. For any given ε > 0, put ε χ = τχΐ3,χ{ε,Οε}. Prom (8.6.13) we
have
(1+С-1)р{8ир^/Я(г)>£}
> PJsup ^/Я(г) > ε\ + P{ inf й/Я(г) < -Cs|
> PJsup \Si\/H(i) > εχ}. (8.6.16)
ν·\οί '
г>2^
By (8.6.4), there exists a 5,0 < δ < 1, such that for every ж > 1
<5Я(2ж) < Я(ж). (8.6.17)

8.6 A further discussion on the complete convergence
237
Hence, we have
{8ир|й|/#(г)>еЛ
oo 2i+1-l
= U U i\Sk\>eiH(k)}
i=j k=2l
oo 2i+1-l oo
^U U {|^1>уЯ(п}и{|^1>уЯ(2Ч}
i=j k=2l i=j
oo 2i+1-l ^
^(J U {|5*-^|>^Я(2*)}
i=j k=2{
= {sup max \Sk - 52*| > -^Я(2г)|,
Combining it with (8.6.16), we obtain
I sup Si/H(i) > εχδ\
i>23
> (1 + «T^^Pf sup max IS* - 52г|/Я(2^) > 2еЛ .(8.6.18)
M>j 2*<fc<2*+1 J
Denote
Bi = < max rr/ .4 > 2ει >,
г l2*<fc<2*+i Я(2·) - Μ'
Аг= П β?>
Z=i+1
*Sfc ~~ ^2*
о* f Dfc ~ ^2* ^ о
Ρ7· = ^ max rr/ .4 > 2εχ
l2i<fc<2^+1-2^-1 Я(2г) ~
It is easy to check that any mixing sequence obeys the 0-1 law. Since
{Bi,i.o.} G f)™=1a(Xk,k > n), wehaveP(P;,i.o.) = lorO. If P(Pi5i.o.) =
1, then
о о oo
PJsup max * N2' > 2εΛ = P([ \ βλ = 1, j = l,2,···.
li>i 2*<fc<2i+i #(2l) - Μ 1.4 J

238
Chapter 8 Laws of Large Numbers and Comolete Convergence
By this equality, (8.6.18) and the property of Mf, we have
oo 2^-1
S(^tf,e0)>E2~J'( Σ ™/>Η
j=2 n=2J~1
2J~1~1 s-
-2 ]T n^(ra))p{sup —^ >ε0}
oo 2^-1 2J'-1-!
>с]Г(2^' ^ ш/>(п) - 2-ϋ-1) ]Γ ш/>Н)
j =2 n=2i~1 n=2J-2
= 6^(2"^ Σ ηψ(η) --ψ(1)) = oo. (8.6.19)
n=2N~1
In fact, for any given Μ > 0 there exists an Λ/ο, when TV > No, η >
2No~1, we have ηψ(η) > 2M, so that 2'N Σ^^ n^(n) - M' (8·6·19)
contradicts with (8.6.8х). Therefore we have
Р(Д, i.o.) = 0,
i.e. Pi U/^i+i ^) —► 0, as г —► 00, so that P(^) —> 1 as г —> oo.
Combining it with (8.6.17), stationarity and the definition of /o-mixing,
for large j we have
PJsup max TT/tx·?* > 2εЛ
1г>^ 2-<fc<2-+i Я(2г) ~ J
00
t=i
00 00 00
> P{\jBiAi} = Y,P{BiAi) >Y^P{B*Ai)
i=j i=j i=j
00
> E{*W)p(*) - Μ2'-1)}
00
> с-£{Р(В*) - ср(?-*)}
00 л
> CY,{P{ max2 5fc > -|itf (2*-1)} - cpi?-1)}. (8.6.20)
Since ηψ{η) > [η/2]ψ([η/2]), Σ%~ι ηψ(η) > Σ%-2~λ ηφ(η), combining

8.6 A further discussion on the complete convergence
239
PJsup max 7777т- > εχδ]
it with (8.6.18), (8.6.20) and (8.6.12) for any given ε > 0, we have
>CE2~'( Σ ^Η-2 Σ гц&(п))
j=2 n=2^~1 n=2J~2
■7>j 2i<k<2i+1 H(k)
oo 2J'-1 2*-1-!
^cE2-i( Σ «νΉ-2 Σ η^(«))
3=2 n=2i~1 n=2i~2
С Sk — Sni Ι
• P{ sup max TT. .4 > 2εχ >
Ii>j2i<fc<2<+1 Я(2г) "Μ
oo i 2^-1
>cEE(2_i Σ "*w
i=2j=2 n=2^~1
-2-ϋ-1) jj~ пф(п))Р(В*А{)
n=2i~2
oo 2i~1-l
>CE2~* Σ ηψ(η)Ρ(Β*Α{)
t=2 п=2{-2
00 2i~1-l 2
>^Σ Σ ^(^(Р^тах^^-^-Ж^-1)}-^-1))
г=2п=2^-2 -
oo
> с Σ 1>(п)Р{тах Sk > -γ Η (η)} - с £ ψ(η)ρ(η)
2£ι
П=1 -_;.- η_^
οο
= cMty, Я, 2εχ/β) - с Σ ψ(η)ρ(η).
η=1
Lemma 8.6.4 is proved.
Theorem 8.6.1 follows from Lemmas 8.6.1 — 3.6.4.
Remark 8.6.1. The condition (8.6.13) is only used in the proof of
Lemma 8.6.4. It can be removed when one considers the convergence of
two-sided tail probability series.
Further discussion of Prohorov's problem for a sequence of independent
random variables were given by Su (1989) and Shao (1991), etc.

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Chapter 9 Strong Approximations
The law of the iterated logarithm and the strong invariance principle
for a sequence of mixing random variables have been discussed by some
mathematicians in the sixties. Applying Strassen's martingale embedding
method to strong approximations for partial sums of dependent random
variables by a Wiener process, Philipp and Stout (1975) established the
almost sure invariance principle for α-mixing sequences, (^-mixing sequences,
lacunary trigonometric series, a class of Gaussian sequences and additive
functional of Markov chains. The results for strictly stationary (^-mixing
and /o-mixing sequences have been improved by Berkes and Philipp (1979),
Dabrowski (1982) and Bradley (1985) respectively. These results are
improved essentially by Lu and Shao. In Shao and Lu (1987), they got
a better approximation order for a (^-mixing sequence. This will be
introduced in Section 9.1. For a sequence of stationary (^-mixing random
variables {Xn, η > 1} with EXX = 0, EX\ < oo and ££°=1 φχΙ2{η) < oo,
Heyde and Scott (1973) first gave that the order of strong approximation
is o((n log log n)1/2). Shao (1989a, 1993b) improved this result and gave
that the same order of strong approximations for (^-mixing and /o-mixing
sequences when the rate of mixing is 0((logn)~~1-e). These results imply
the law of the iterated logarithm. When the (2 + <5)-th moment is finite,
he gives the further results which imply the Chung law of the iterated
logarithm. We shall discuss these results for the (^-mixing and /o-mixing cases
in Section 9.2. Shao and Lu(l987) and Shao (1989a) have also studied the
strong approximations for an α-mixing sequence. These will be given in
Section 9.3.
9.1 Strong approximations for a (^-mixing sequence
Let {Xn, η > 1} be a sequence of (^-mixing random variables with
EXn = 0. Put Sn = Σ%=ι Xk, S(t) — S[t] (t > 0)· In this section, we first
discuss how fast is the rate of strong approximations for a sequence of φ-
mixing random variables when applying Strassen's martingale embedding

242
Chapter 9 Strong Approximations
method. The rate of strong approximations gotten by Shao and Lu (1987)
approaches the ideal order (9((nloglogn)1/4(logn)1/2) for a sequence of
i.i.d.r.v.'s. The following theorem is proved.
Theorem 9.1.1. (Shao, Lu 1987) Let {Xn-, η > 1} be a sequence of
φ-mixing random variables with EXn = 0. Suppose that
(i) σΐ = ESl > Cn for some С > 0,
(ii) supn EX* < oo,
(Hi) φ(η) = 0(l/n).
Then without changing the distribution of {S(t), t > 0}, we can redefine
the process {S(t), t > 0} on a richer probability space together with a
standard Wiener process {W(t), t > 0} such that for any ε > 0
S(t) - W{oj) = 0(at1/2(logat)9/4+e) a.s. (9.1.1)
as t —> oo, where σ\ = aL.
Prom Theorem 9.1.1 we have the following corollary immediately.
Corollary 9.1.1. Let {Xn, η > 1} be a sequence of stationary φ-
mixing random variables with EX\ = 0. If
oo
σ2 = EX\ + 2J2 EXiXk (9.1.2)
fc=2
is absolutely convergent, by assuming σ = 1 without loss of generality, then
under the conditions (ii) and (Hi), we have
S(t) - W(t) = 0(*1/4(logtf/4+£) a.s. (9.1.3)
In order to prove Theorem 9.1.1, we first give a fundamental
proposition. The proof of the proposition points out how to use the Strassen's
martingale embedding method for a mixing sequence to get the results of
strong approximations.
Let To — {0? Щ and {^, η > 0} be a sequence of non-decreasing
σ-fields. Assume that Xn is ^-measurable, η = 1, 2, · · ·. Define
2n
Yn = 2^{Е(Хп+к\ Pn) ~ E(Xn+k\Fn-\)}
fc=0
= Xn + un - un-i - vn (9.1.4)

9.1 Strong approximations for a ^-mixing sequence
243
for every η > 1, where
2n 2n
un = Y^E{Xn+k\Tn), vn= JT E(Xn+k\fn-i). (9.1.5)
k=l k=2n~1
{Yn, Tn, η > 1} is a martingale difference sequence.
Proposition 9.1.1. Let {Χη? η > 1} 6e α sequence of random
variables with EXn = 0, supn Ε\Χη\2+δ < oo (0 < δ < 2). Le* JFn =
σ{Χ&, 1 < к < η} be a natural σ-field sequence. If the following
conditions are satisfied:
(α) σΐ = ESl > С η for some С > 0,
(b) K||2+, = O(l), Σ~=ι bnh+s < oo,
(c) for some λ > 0
E\ £ (П2 - ΕΥξψ^2 = 0(n(logn)A), (9.1.6)
m<k<m+n
then without "changing the distribution of {S(t), t > 0}, we can redefine
the process {S(t), t > 0} on a richer probability space together with a
standard Wiener process {W(tf), t > 0} such that for any ε > 0
S(t) - W{a2t) = о(а2/(2+г)(1о§а()1+£+(1+А)/(2+г)) a.s. (9.1.7)
as t —> oo. Particularly7 for δ = 2 we have
S(t) - W{a2t) = o(alt'2{\ogatY^IA+£) a.s. (9.1.8)
Proof. 1) We first prove that for each ε > 0,
S(t) - Y^Yk = O(^/(2+^(log0(1+£)/(2+5)) a.s. (9.1.9)
k<t
In fact, noting that
5(ί)-ΣΥ* = Στ;*-Μ[Φ
k<t k<t
using condition (b) and the Borel-Cantelli lemma, we conclude that (9.1.9)
holds.
We now apply a martingale version bf the Skorohod-Strassen
representation theorem. There exists 'a probability space, on which a standard

244
Chapter 9 Strong Approximations
Wiener process and a sequence of non-negative stopping time Tn are
defined such that
ΜΣ2ί).»^1} and {Σ^^1}
have the same distribution. Hence without loss of generality, on this new
probability space we can redefine Yn by
j<n j<n
and keep the same notation. Now, write Tn = cr{W{^-<k Tj)\ 1 < к < η},
and Qn = a{W{t); 0 < t < Y^j<n Tj}. It is easy to see that Tn С C/n, Tn is
^-measurable and for every η > 1,
E{Tn\Qn-{) = E(Y^\Un^) = E{YZ\Fn_x) a.s. (9.1.10)
Moreover, for each 1 < ρ < 2,
Е\Тп\р < E\Yn\2?.
2) Write
S(t) - W(a2) = S(t) - ^П + w(Er*) - W{a2t). (9.1.11)
k<t k<t
In order to estimate the second difference of the right hand side of
(9.1.11), we have to estimate ]Cfc<t^fc ~~ at· Write
Er*-*i
k<t
= Е{г*-я(здк_1)}
k<t
+ Σ{Ε(Υέ\η-ι) - Yi) + {Ση2 - <??}■ (9-1-12)
k<b k<t
We shall show that
Y^Tk - σ\ = 0(а,4/(2+5)(1оёа,)1+£+2(1+А)/(2+5)) a.s. (9.1.13)
k<t
Put Rj = Yf - E(Y^\Tj-i). Then {Rj,Fj} is a martingale difference
sequence. We obtain from condition (b) that
£|Д |(2+*)/2 < 16£|^.|2+* = 0(^β

9.1 Strong approximations for a ^-mixing sequence
245
Hence by the fundamental theorem on the martingale (cf. Chow 1965), we
have
£ Д*.= O(^2/(2+^(log0(2+£)/(2+5)) a.s. (9.1.14)
k<t
Similarly, we have
£(Tfc - ВДйк-х)) = 0(i2/(2+fi)(logi)(2+£)/(2+6)) a.s. (9.1.15)
k<t
For the third term of the right hand side of (9.1.12), by the condition (c),
Moricz's theorem and the Borel-Cantelli lemma, it is easy to show
Σ{Υ* - EY%) = O(^2/(2+^(log01+£+2(1+A)/(2+5)) a.s. (9.1.16)
k<t
Noting that {Yk^k} is a martingale difference sequence, from the
conditions (a), (b) and the Schwarz inequality, we obtain
k<t k<t
= 2E(£xk) («* - Σν^> + E{ut - Συ*)2
k<t k<t k<t
= 0(at). (9.1.17)
Hence by (9.1.16), (9.1.17), and condition (a), we have
£n2 - σ2 = 0(at4/(2+6)(logat)1+£+2(1+A)/(2+fi)) a.s. (9.1.18)
fc<t
Equality (9.1.13) follows from (9.1.14), (9.1.15) and (9.1.18).
We now conclude
W(^Tk) - W{a2t) = 0(σί2/(2+5)(1οβσί)1+ε+(1+λ)/(2+5)) a.s.
k<t
by (9.1.13) and the proof of Theorem 3.2B in Hanson and Russo (1983).
Combining (9.1.11) with (9.1.9) yields the proposition.
Proof of Theorem 9.1.1. We now verify that the conditions
(b) and (c) of proposition 9.1.1 are satisfied. By Lemma 1.2.8 and Lemma

246
Chapter 9 Strong Approximations
2.2.8, we have
hnWlXs = ΕΚ\2+δ = E(un(sgnun)\un\1+6)
= E(Xn+1\un\1+6sgnun) + ΣΕ[ ]Γ Xn+k\un\1+6sgnunJ
г=0 fc=24l
= o(Kiia + Σ ^+6)/{2+^)2Пип\\1Х66)
i=0
= o(\\un\\nss),
which implies that
\\un\\2+6 = 0(l). (9.1.19)
Similarly, we have
\Ы\2+6 = 0(φ^+δ^2+δ\2η-1)2^-1^2).
Hence
oo
Σ \\νη\\2+δ < oo. (9.1.20)
71=1
Therefore condition (b) is satisfied.
For condition (c), put
Tm{n)= Σ (Ук-EYi), rn = sup||Tm(n)||2. (9.1.21)
m<k<m+n
It is easy to obtain
||Гт(п)||2 < ||Tm([n/2]) + Tm+[il/2]+/([n/2])||2 +·2η + 2rx, (9.1.22)
where / = [2n(log(2n))~2~e]. Note that
E(Tm([n/2]) + Tm+[n/2]+l([n/2}))2
< 2rfn/2] + 2E{Tm([n/2])Tm+[n/2]+l([n/2})}. (9.1.23)
By an elementary calculation, from (9.1.4) and Lemma 2.2.8 we obtain
ETm([n/2])Tm+[n/2]+l([n/2})
<2^/2(0||Tm([n/2])||2|^xJ
к
+ iirn([n/2])i|2(i|u^ii4 + \\un2\u + Σ\ыи) Ι Σχ^|
к к
+ цгт([п/2])||2 (\\uNl Hi + \\uN2\\\ + Σ \ы\1),

9.2 Strong approximations for a p-mixing sequence
247
where J2k is extended over m + [n/2] + I < к <m-\- 2 [n/2] + / and
N2 = m + [n/2] + /, Ni = m + 2[ra/2] + l.
Using Lemma 2.2.8, (9.1.12), (9.1.13) and conditions (i) , (ii), we
conclude that there exists a constant С such that for each ra,n > 1,
ETm([n/2])Tm+[n/2]+l([n/2}) < CV([n/2]) n1'2 (logn)1+e. (9.1.24)
Prom (9.1.22), (9.1.23) and (9.1.24), we obtain
||Tm(n)||2 < 21/2r([n/2]) + Cn1'2 (logn)1+e
+r([2n(logn)-2-£]). (9.1.25)
Finally, from (9.1.25) and by induction we have
Tn<C0n^2(logn)2+£,
with Co = max(exp(22/e),2C). This shows that condition (c) holds for
λ = 4 + ε, and Theorem 9.1.1 follows.
Remark 9.1.1. Let {Xn\ η > 1} be a ^-mixing sequence with
EXn = 0. Suppose that condition (i) of Theorem 9.1.1 is satisfied and for
some 0 < δ < 2,
s\npE\Xn\2+s < oo
Π
and φ{η) goes to zero with the polynomial rate. Shao and Lu (1987) have
also given the following results:
1) If 0 < δ < 2 and φ(η) = 0(n~a) for some a > 1, then
S(t) - W{a2t) = 0(а,2/(2+5)(1оёа,)1+£+(1+А)/(2+5)) a.s.
for any ε > 0, where λ = 2 log3/log <9_1,6> = 1 - 2(a - l)/(a(2 + £)) > 0.
2) If 0 < δ < 2 and φ(η) = 0(n-a), (2 + £)/(2(l + δ)) < a < 1, then
S(i) _ W{a2t) = 0(σΙ-αδ/{2+δ)+ε) a.s.
for any ε > 0.
9.2 Strong approximation for a p-mixing sequence
In the introduction of this chapter, we have mentioned an interesting
problem: what the rate of strong approximation of partial sum process S(t)

248
Chapter 9 Strong Approximations
for a mixing sequence by a Wiener process is when we only assume that
supn EX2 < oo and φ{η) = <3((logn) α), a > 1 (or p(n) = 0((logn)-a), a
> 1) ? In this section, we deal with a p-mixing sequence, and the strong
approximation order 0((n log log n)1/2) is given. With this result the law
of the iterated logarithm holds true for this p-mixing sequence. In order to
get the Chung law of iterated logarithm for a /o-mixing sequence, we need
the assumption supn Ε\Χη\2+δ < oo.
Theorem 9.2.1.(Shao 1989a, 1993b) Let {Xn, η > 1} be a strictly
stationary p-mixing sequence with EX\ = 0, EX\ < oo. Assume that
(i) σ2 = ES„ —> oo as η —> oo,
(it) p(n) = 0((logn)~~1-e') for some ε' > 0.
Then without changing the distribution of {S(t), t > 0}, we can redefine
the process {S(t), t > 0} on a richer probability space together with the
standard Wiener process {W(tf), t > 0} such that
S(t) - W{a2t) = o(at(loglog*)1/2) a.s. (9.2.1)
as t —> oo.
Theorem 9.2.2.(Shao 1989a, 1993b) Let {Xn, η > 1} be a strictly
stationary φ-mixing sequence with EX\ = 0,EX2 < oo. Assume that
(i/ σ2 —> oo as η —> oo;
(η) Σ^°=ι ^1/2(2η) < oo. Then (9.2.1) is also true.
With the help of the law of the iterated logarithm for a Wiener process,
from the previous theorems we have the following corollary.
Corollary 9.2.1. Under the conditions of Theorem 9.2.1 or 9.2.2, we
have
limsuplSnl/W^logloga2 = 1 a.s. (9.2.2)
n—юо
The following strong approximation theorem implies the Chung law of
the iterated logarithm.
Theorem 9.2.3.(Shao 1989a, 1993b) Let {Xn,n > 1} be a strictly
stationary p-mixing sequence with EX\ = 0, Ε\Χι\2+δ < oo for some
δ > 0. Assume that
(i) σ2 —> oo, as η —> oo/
(ii) p(n) = 0((logn)~r) for some r > 1/2.

9.2 Strong approximations for a p-mixing sequence
249
Then for any 0 < θ < r/2 — 1/4, we have
S(t) - W{a2t) = o{at{\ogt)-°) a.s. (9.2.3)
Corollary 9.2.2. Under the conditons of Theorem 9.2.3, we have
(9.2.2) and
liminff %-t—) max|SJ = l a.s. (9.2.4)
ЛГ-+00 V π2σίΓ l i<n<N' '
Remark 9.2.1. The results of Theorem 9.2.3 and Corollary 9.2.2
hold true for a stationary (^-mixing sequence with EX\ = 0, £?|Xi|2+<5 <
oo, σ\ -► oo and φ(η) = 0((logn)~r) for some r > (2 + <5)/(2(l + 5)).
Remark 9.2.2. The stationarity condition in these theorems and
corollaries can been replaced by the identical distribution condition.
We point out in passing that the results in this section improve those of
Bradley(1985), Dabrowski(l982) and Theorem 4 of Berkes and Philipp(1979).
In the proof of these theorems , we shall use both the Bernstein divided-
section method and the Strassen martingale embedding method. Put
Xk = XkI{X2k >k)- ЁХк1{Х2к > k), (9.2.5)
Xk = XkI{X2k <k)- EXkI{X2k < k). (9.2.6)
Let
S(n) = J2xk, S(n) =Y,Xk,
fc=l fc=l
k+n k+n
Sk(n) = Σ Xt Sk(n) = £ Xi.
i=k+l i=k+l
We shall first prove
~S(n) = oin1'2) a.s. (9.2.7)
Define blocks of integers i/χ, /χ, #2, h,' * * by requiring that Hk contains hk
and Ik contains ik consecutive integers and that there are no gaps between
consecutive blocks, where
hk = Card Hk = [aka~l exp(A;a)],
ik = Card Ik = [ak*-1 exp(A:a/2)] (9.2.8)

250
Chapter 9 Strong Approximations
and 0 < a < 1 will be chosen later on. Put
Nk := Σ CaidiHj U Ij) ~ exp(A:a),
j<k
uk := ]T Xh vk := Σ Xi>
jeHk jeh
Ck := uk - E(uk\Tk-i) where .Γ* = σ{Χ^ι < Nk-X + hk),
mn := {k : η e HkU Ik} - 1 ~ (logn)1/a.
It is clear that
Sn = S(n) + S(n). (9.2.9)
t=l г=1
77ln П
+ Σ> + Σ *··· (9·2·10)
t=l t=JVmn + l
In order to prove Theorem 9.2.1, by this two representations , we shall
show at first that one needs only to check (9.2.7) and to show that the
following relations hold true:
Y^Vi = o(exp(na/2)) a.s. (9.2.11)
г=1
3
max V Xi = o(exp(na/2)) a.s. (9.2.12)
^<J<Nn+i\i=^+1 I
η
5^JS(iXi|^-i) = o(exp(na/2)) a.s. (9.2.13)
г=1
oo
Σ e~(1+5l)ia£;|6|2(1+5l) < oo for some δχ G (0,1) (9.2.14)
t=l
η
Σ(£?(ί1?|^_1)-^?)=ο(βχρ(ηβ)) a.s. (9.2.15)
г=1
5>&2-£S*=o(n) a.s. (9.2.16)
г=1

9.2 Strong approximations for a p-mixing sequence 251
Proof of Theorem 9.2.1.
Assume that (9.2.7) and (9.2.11)-(9.2.13) hold. Denote σ2η = Σ£α ЩЬ
Then
S(n) - Σ & = oin1'2) a.s. (9.2.17)
t=l
By the Strassen martingale embedding method, for the martingale
difference sequence {ffc, T^k > 1} there exist the stopping times {Tn,n > 1}
such that
{w(£K),n>i}U:{^i,n>i}.
t=l t=l
Redefine
η n—1
& = W(Ti), tn = w(Y/Ti)-w(Y/Ti), n>2.
г=1 г=1
As in (9.1.10), we have
E(Tn\ Gn-г) = E{en\ Gn-ι) = Ε(£\ Τη-ι)·
Write
and
Σ&-^(^) = ^(Σ Τ0~^(Σ ΕΤ) (9·2·18)
г=1 г<тп г<тп
= Σ(τ<-£?№|&-ι))
г=1
η
г=1
η
+ Σ(£«?ι *-0 - £) + Σ«? - *# )· (9·2·19)
t=l t=l
By a martingale result of Chow (1965) and (9.2.14), the series
oo
Σ>~η4η<°° a.s.
t=l
implies that the series
oo
г=1

252
Chapter 9 Strong Approximations
is also almost surely convergent. It follows from the Kronecker lemma that
Σ(ί? - *#) = о(ехрЮ) a.s.
г<п
By the same way, we have Σί<η(Τί — E{Ti\Qi-i)) = o(exp(na)). Hence, in
combination with (9.2.15), we have
η
Y£Ti - ETi) = o(exp(na)) a.s.
t=l
Using the Hanson-Russo Theorem for the lag increments of a Wiener
process (see Hanson and Russo 1983 Theorem 3.2.B) and (9.2.18), we get
E^"W(*n) = o((nlogtogn)1/2) a.s. (9.2.20)
t=l
Similarly, from (9.2.16), we have W(a2)-W(a2) = o((n log log n)1/2) a.s.
Combining it with (9.2.20) and (9.2.17), the conclusion of Theorem 9.2.1
follows.
Now in order to complete the proof of Theorem 9.2.1, it is sufficient to
prove (9.2.7), (9.2.11) —(9.2.16). They will be given by a series of lemmas
in the following. At first, we give three lemmas for any sequence of random
variables. Lemma 9.2.1 implies (9.2.7).
Lemma 9.2.1. Let {Xn} be a sequence of strictly stationary random
variables7 EX\ < oo. Then
J2{\Xi\I(\Xi\ > г1'2) + E\Xi\H\Xi\ > i1'2)} = oin1'2) a.s. (9.2.21)
t=l
Proof. We have
oo
J\-^E\Xi\I{Xf > г)
oo j
<EEi-1/2E\Xi\lU-KXf<J)
3=1i=l
oo
Ki^U-l^ElXrllU-KX^Kj)
OO
<ΑΣΕΧΐΐ{3-ΚΧΐ<ί)
3=1
= 4EXf < oo.

9.2 Strong approximations for a p-mixing sequence 253
Therefore T£Lii~l,2\Xi\I{Xi > г) < oo a.s., then (9.2.21) follows from
the Kronecker lemma.
Lemma 9.2.2. If E\X\ < oo, then for any 0<6<1,ε>0
oo
Σ^β-^ΕΙΧ^+'Ι^ΧΙ < ekb) < oo.
k=l
Proof. It is easy to see that the left hand side of the above
inequality equals to
oo
Σ^β-^ΕΙΧ^Ι^Χ] < 1)
fc=i
к
+ У£кь-1е-£кЬ^/Е\Х\1+Ч(е«-1? < \X\ < /)
k=l 1=1
oo
<c + c ΣΕ\Χ\ι+4(β^ < \X\ < elb)e-£lb
1=1
< cE\X\ < oo.
Lemma 9.2.3. Let {cn, η > 1} be a sequence of nonincreasing positive
numbers. Then, for any real sequence {ηη, η > 1}, we have
г г
max Ι Σ »?j I Q < 2 max I ^ CjVj I. (9.2.22)
1<г<п I 7~"-r I 1<г<п I 7—* I
3=1 3=1
Proof. Denote D{ — Σ)=ι cjVj- We have
щ = (Dj-Dj-J/cj = Σ(γ " 7~YDi ~D3-^
i=i Ci Ci~1
where 1/co = 0. Therefore

254
Chapter 9 Strong Approximations
It follows that
к
max си У^г?7 < max max \Dk — A-i
i<k<n fclf-i 3\ ~ i<k<ni<i<k]
3=1
к
< 2 max I VcjtJ. (9.2.23)
i<k<n I f—ί I
The Lemma is proved.
Lemma 9.2.4. Let {ХП5 ^i > 1} be α p-mixing sequence with EXn =
0, EX\ < oo. Le£ Uk,Vk be as above. Denote ик(п) = Y^i=k+1Ui, Vk(n) =
Y^i=k+iVi- Suppose that
oo
Σ P(2") < oo.
71=1
TTien 2/iere exists a constant с = c(p(·)) such that for any к > 0,n > 1
k+n
Еи2к(п)<с( Σ £«i)' (9·2·24)
i=k+l
Evl(n)<c( ζ Ευή. (9.2.25)
i=k+l
Proof. We only give the proof of (9.2.24). It is obvious for η = 1.
For η > 2, denote ri\ — η — [ra/2], ri2 = [ra/2]. Then by the definition of
p(·) we have
St4(ra) = Eul(ni) + s*4+ni(ra2) + 2Euk(n1)uk+ni(n2)
< (ЕиЦщ) + Bul+ni(n2))(l + ρ(ιηι)). (9.2.26)
Let ci = l,cn = 0^(1 + /9(гП1))(п > 2). It is easy to see that cn is
non-decreasing. It follows from (9.2.26) that for any к > 0, η > 1
k+r,
£?ul(n)<<v,( Σ *Ч2)·
г=&+1
Now, we prove that {c™} is a bounded sequence. Note that
C2m = C2m-l(l +/9(i2m-l))
771— 1
< c2m-i exp(p(i2m-i)) < expj ]П p(i2i)}·

9.2 Strong approximations for a /э-mixing sequence 255
From the definition (9.2.8) of г^, for any large j, we have
p{i23)<p{22ia),
and further
m—1 m—1 m—1
Σ^2,)<Σρ(22'α)<Σ^)<°°
j=o j=o j=o
by monotoncity of p(·). This proves C2m < с < oo . It follows from the
monotoncity of Cn that \cn\ is bounded. The lemma is proved.
The following lemma gives (9.2.11).
Lemma 9.2.5. Let {Xn,n > 1} be as in Theorem 9.2.1, we have
η
]Γ Vj = o(exp(na/3)) a.s. (9.2.27)
Proof. Using Lemma 2.2.5 with q = 2 and condition (ii), we have
η
E(S~]vj)2 ^ cn т&х Еу]
. ^ 1 <C 7 <Cn
<cnaexp(na/2).
It follows from the Borel-Cantelli lemma that (9.2.27) holds true.
The following lemma gives (9.2.12).
Lemma 9.2.6. Assume that the conditions of Theorem 9.2.1 are
satisfied and
0<α<ε'/(8 + ε'). (9.2.28)
Then
j
max \ У^ Xi\=o(exp(ka/2)) a.s. (9.2.29)
Nk<j<Nk+1\.^k I
Proof. By the Borel-Cantelli lemma, we need only to prove that
for any ε > 0
oo j;
У P\ max У ΧΛ > sexp(fca/2)) < oo. (9.2.30)
έί ^Nk<3<Nk+1\^k I" П
Put dfc = Nk+1 - Nk ~ afca~x exp(/ca), and let
β = fc-ette+O/e' exp(ifce/2), m = [Ara(16+£'>/e' eXp(fce)].

256
Chapter 9 Strong Approximations
For any ε > 0, we have
A&mEX^I(\X1\ >B)< eexp(ka/2) (9.2.31)
В
for large k. It follows from Lemma 4.3.2 that
3
P{„^jE^>e«P<*72)}
<c{exp(-^r)(48+£')/8
+ 4bg2+£'/4(24)^|Xi|2+e'/4/(l^i| < B)
+ dkE\X1\2+*''4I(\Xl\<Nk+1))
+ exp(-ka)dk(l + p2(m) log4[<4/m])}
< c{fc-(l-«)(8+e')/8 + k-l-a
+ ka-1exp(-e'ka/8)E\X1\2+£'/4I(\Xl\2 < 2efc°)}
+ c^ka-l-2a(l+e')log4ky
By Lemma 9.2.2 and (9.2.28), (9.2.30) holds true.
The following lemma gives (9.2.13).
Lemma 9.2.7. Suppose that condition (ii) of Theorem 9.2.1 is satisfied.
Then we have
η
Σ E(uk\ J^_i) = о (ехр (na/2)) a.s. (9.2.32)
fc=i
Proof. We first prove that for any к > 0, η > 1 and a sequence of
real numbers {cn}
EG2k(n) <c{J2 P2d(J/2)) ή En)} log2(2n) (9.2.33)
j=k+i
where Gk(n) = Σ^%+1 Cj E(uj\ Tj-i) and i(x) is the linear interpolation
function of ik· From (9.2.24) , there exists a constant d such that for any
* >0,n> 1
k-\-TL K-\-TL
E{ Σ (Чщ)2 < c'( £ <$Ε*ή· (9·2·34)
i=k+l i=k+l

9.2 Strong approximations for a p-mixing sequence 257
Put с = 100c' log-2 (3/2). Apply the induction on n. When η = 1
EG2(l)=c2k+1E(E(uk+1\Tk))2
<4+1р(гк)\\ик+1\\2\\Е(ик+1\^к)\\2
i.e.
EG&l) < ck+1p2(ik)Eu2k+v
This proves (9.2.33) for η = 1. Suppose that (9.2.33) holds true for any
integer less than n. We prove that (9.2.33) remains true for n. Denote
щ = η — [n/2], ri2 = [n/2]. We have
EG2k(n) = EGl(m) + EG2k+ni(n2) + 2EGk{nx)Gk+ni{n2)
= EGl(n1) + EGl+ni(n2) + 2EGk(n1)( £ cJuj)
j=fc+ni+l
<ЕСЦщ) + ЕС1+П1(п2)
к+п
+ 2p(ifc+ni)|| Gk(ni)\\2 J Σ . cj uj
j=k+ni+l
By (9.2.34) and the assumption of induction, we have
fc+n
EG2(n)<c{ £ ρ2(i(j/2))c2En2} log2(2щ)
+ 2\/^p(iA;+ni){ ^ ή Ей]}
j=k+ni+l
.{ Σ p2(i(j/2))^JBu?} Iog(2m)
<(c(log2n1)2 + x^log(2n1)){ Σ p2(i(j/2))c2£u2}
j=fc+i
<^{ Σ p\i{3m)c)Eu2}\og2{2n).
j=k+i
(9.2.33) is proved.

258 Chapter 9 Strong Approximations
It follows from (9.2.33), ΣΪΖι p(20 < °° and Lemma 2.2.2 that
η
P{\ ^(«il^-OI > εβχρ(ηα/2)}
г=1
<ce-na{J2r2aEu>}log2(2n)
г=1
<ce-"°{^ra-1ei<1}log2(2n)
i=l
< cn_2olog2(2n).
Let η^ = [Α;χ/α]. By the Borel-Cantelli lemma, we have
J2E(ui\Ti-1) = o(exp(nak/2)) a.s. (9.2.35)
г=1
Moreover from Lemma 9.2.3, Lemma 2.2.2, Lemma 4.1.2 and (9.2.33), we
have
P{ max Ι Σ e-ja/2E(ui\Ti^)\ > ε}
l=nk
j
< P< max
^rik<j<nk+1 I /—' I J
l=nk
nk + l
<c{ Σ r2ae-ja^}log4(nfc+1-nfc)
< cAT2(logA;)4,
which implies that
j
max
Σ е~Г/2£ЫЯ-1) -> 0 a.s. (9.2.36)
l=nk
Thus (9.2.32) follows from (9.2.35) and (9.2.36).
The following lemma gives (9.2.14).
Lemma 9.2.8. Suppose that a satisfies (9.2.28), and condition (ii) of
Theorem 9.2.1 is satisfied. Then for δ\ = ε'/4, we have
Σ 6-(ι+*ι)*α Ε\&\2+2δ> < oo. (9.2.37)
fc=i

9.2 Strong approximations for a p-mixing sequence
259
Proof. From Lemma 2.2.5, Lemma 9.2.2 and condition (ii), we
have
Е\Ы2+261 <сЕ\ик\2^
+ ka-1exp(ka)E\X1\2+26lI(Xf < 2exp(A:a))}.
Then, by 0 < a < ε'/(8 + ε') and Lemma 9.2.2, (9.2.37) is proved.
The following lemma gives (9.2.15).
Lemma 9.2.9. Suppose that a satisfies (9.2.28) and condition (ii) of
Theorem 9.2.1 is satisfied. Then
η
Σ(β(ί?Ι·^-ι) ~ ΕΦ = o(exp(na)) a.s. (9.2.38)
i=i
Proof.
|E(jB(f?i^!)-jBf?)|
3=1
<\Σ{Ε{η)\Τ^)-Εη))\
3 = 1
+ Σ{Ε\ηό\ Ъ-г) + Е{Е{щ\Т^))2). (9.2.39)
3 = 1
We first prove that
£(£2(и,-|^-0^ a.s. (9.2.40)
j=i
In fact, by the definition of ρ(·) , we have
E{E{uj\Tj-X))2 = EiujEiujlFj-t))
<р(Ч-1)\Ы\2\\ЕЫЪ-г)\\2.
Therefore
Е{Е(ч\Ъ-г))2 <р2(г^)Еи2 < cj-W^V,

260 Chapter 9 Strong Approximations
and further
oo oo
ΣΕ{Ε{ηό\Τ^)γ/β1α < с ΣΓι-«(ΐ+^) < oo.
3=1 3=1
By the Kronecker lemma, (9.2.40) holds true.
Secondly, we prove
η
Σ(Ε(η)\ Tj-ι) - Ей)) = о (exp(na)) a.s. (9.2.41)
i=i
Denote щ — и21(\щ\ < е%а/2). It is easy to see that
\Σ{Ε{η}\Τ^)-Εη})\
i=l
η
г=1
+ jr(E(ull(\ui\>eia/*)\?i-1))
г=1
+ Eu2iI{\ui\>eial2)). (9.2.42)
It follows from Lemma 2.2.5 and Lemma 9.2.2 that
oo
Y,E{E{u)H\ui\ > *а'2)\Ъ-1))1*а
oo
= J2e-^Eup(\Uj\>e^)
3=1
OO
< с^е-(х+6^аЕ\щ\2+6*
3=1
OO
< с ^e-(i+*i/2)ia{(j«-ieia)i+*i/2
3=1
+ e^r-^lXxl^1/^2 < 2e'°)}
< OO.
By the Kronecker lemma
Y^{E{u)l{\Uj\ > е^12)\Г^) + Еи)1{\и0\ > еГ**))
3=1
= о (exp(na)) a.s. (9.2.43)

9.2 Strong approximations for a p-mixing sequence 261
On the other hand, by the same way as in the proof of (9.2.33) in Lemma
9.2.7, there exists a constant С such that
£(Σ E№ - Sftl 'i-i)))2 < c (Σ /К*С#/2)) Ей]) log2(2n).
I2
Then by Lemma 2.2.2 and condition (ii), we have
Р{\^Е(Щ - Е(Щ\Т0-г))\> ееп°}
-л
<ce-^\±r2a^£,)e^){\ognf
<
< cn-2fl(1+e') (logn)2.
By the same discussion as in the proof of second half part of Lemma 9.2.7,
it follows that
η
"Y^Eiuj — Euj\Tj-\) — o(exp(na)) a.s.
i=i
Combining it with (9.2.42), (9.2.43) yields (9.2.41). Lemma 9.2.9 is proved.
The following lemma gives (9.2.16).
Lemma 9.2.10. If the conditions of Theorem 9.2.1 are satisfied, we
have
J2Eg-ESl=o(n). (9.2.44)
Proof. Denote Uj — Σ/ея· ^ь Vj = Σι^ι- Χι- We have
2^ = s(Efii + E®i + Σ Хз)
i=i i=i i=N(mn)+i
= ^(Efii) +^(Σ%+ Σ *;)
J=l 3=1 j=N(mn)+l
mn mn η
+ 2ε(ς>)(Σ^+ Σ *;)· (9·2·45)
j=l j=l j=W(m„)+l

262 Chapter 9 Strong Approximations
By Lemma 9.2.4 and the definitions of Nk and ran, it follows that
mn
Ε(Έ^+ Σ xj) =о(ч£щ + (п-мтп))
j=l j=N(mn)+l i=l
0{n(logn)^). (9.2.46)
Write
Ε{Σ^γ = e(J2Uj + ς^ - Uj))2.
3=1 j=l 3=1
Prom Lemma 9.2.4 and Lemma 2.2.2 we obtain
rnn 2 rnn
j=i j=i
3=1
= o(n).
On the other hand,
Σ*«-£(Σ«;)
3=1 3=1
= s(Efc) -*(Σ«>)
j=l j=l
= £(Σ%|^-ι)) +2£(^Uj)^%-|7H).
3=1 3=1 3=1
It follows from (9.2.33) that
/mn \2
Я(Х>(«,-|.Г,·-!))
mn
c(Er2(1+£)a^i)(bgmn)2
j=i
< cn(logn)-2(1+£)(loglogn)2.
77ln
<
Combining it with the above relations yields (9.2.44) .
Now the proof of Theorem 9.2.1 is completed.

9.3 Strong approximations for a α-mixing sequence
263
The proof of Theorem 9.2.2 is the exactly same as that of Theorem
9.2.1, provided Lemma 2.2.10 is used instead of Lemma 2.2.5 and 4.3.2.
The proof of Theorem 9.2.3 is similar to that of Theorem 9.2.1 as well.
We need only to apply the Bernstein divided-section method for {Xn}
immediately. The details are omitted.
9.3 Strong approximations for an α-mixing sequence
Shao and Lu (1987) improved, the strong approximation result of
Philipp and Stout (1975) for an α-mixing sequence, and obtained a better
rate of strong approximation when using the Strassen martingale
embedding method.
Theorem 9.3.1. Let {Xn,n > 1} be an α-mixing sequence with
EXn — 0 and g(x) a function such that g(x)/x2+6 for some 0 < δ < 2 is
increasing to infinity. Denote
\\X\\g = inf{i > 0, Eg(\X\/t) < 1}.
If supn ||Xn||i7 < oo and the following conditions are satisfied
(i) σ\ = Ε Si >Cn, for some С > 0,
OO 1 / \
(it) Σ а(п)ш'1пУ9[-^Ь)) < oo ,
71=1 \ V //
then
S(t) - W{c2t) = 0(a(2/(2+i)(logat2)1+(1+A)/(2+6)) a.s. (9.3.1)
where λ = (log2)/log((2 + δ)/δ) < 1.
Proof. Write
oo
Yn = ^{Е(Хк+п\ Tn) - ВДь+nl .Fn-i)}
= Xn + un - un-i, (9.3.2)
oo
where un — J] E(Xjc-\-n\^rn-i)- Let us verify the conditions of Proposition
k=o
9.1.1. Condition (a) is assumed. To check condition (b), denote
Ukn = E(Xk+n\Fn-l)·
Take f(x) = ж(2+*)/(1+*) in Lemma 1.2.3, we have
Е\икп\2+6 = Е(Хк+пикп\икп\6)
<с^щ)<*(к)1/{2+6)\\Хь+п\\9\\

264
Chapter 9 Strong Approximations
Therefore
lkn||2+* <cimg(l/a(k))a(k)^2+s\
It follows from (ii) that condition (b) of Proposition 9.1.1 is satisfied.
Denote
Tm(n)= Σ (Yk2-EYk%
m<.k<m+n
rn = sup£|Tm(n)|(2+5)/2.
m
We shall show
τη <cn(logn)A (9.3.3)
by induction on n, where
λ = (bg —-—)/ log -^— < 1,
1 < Ms < 2, 0 < δ < 2.
Put
β = δ/(2 + 6), щ = η - [n% n2 = [ηθ].
FVom an lementary inequality
|1 + ж|р < l+px + Cp\x\p, fori <p<2, 1 < Cp < 2,
we have
£|Тт(п)|^ < £|Тт(щ)|^ + MsE\Tm+ni{n2)f^
'2 + 6
+ (—-)я|Гт(щ)|* |TTO+ni(n2)|
< rni + M6rn2 + (—_)£7(|гта(т)|з
* {( Σ XJ +^Wi -WJV2)
7n+n 2
~ E{ Σ -Xj·+«Ni - «iv2)})
i=7V2
=: τηι + М6тП2 + -^— /, (9.3.4)

9.3 Strong approximations for a α-mixing sequence 265
where
JVi = m + η + 1, N2 = m + nx + 1.
Note that
m+n
I = E\Tm(ni)\l Σ(Χ]-ΕΧ])
j=N2
+ E\Tm(ni)\^((uNl - uN2)2 - E(uNl - uN2)2)
m+n
[uNl -uN2)
J=N2
m+n
~Ε(Σ Xj)(UNi -UN2)}
j=N2
+ 2E\Tm(ni)\2 22 (Xj+N2-lXi+j+N2-l
- EXj+N2-l^i+j+N2-l)
=: h+I2 + 2/з + 2/4. (9.3.5)
From Lemma 1.2.3 ( take g\{x2) = g(x),f(x) — χ(2+δΜδ) and condition
(ii), we have
00 1 ..
h < с(Е\Тт(П1)^)Ш^пу91(^т/(--)а(к)
k=0
00
<с{Е\Тт{П1)\^)Ш.
By the Holder inequality and condition (b), we have
h < с(Е\Тт(щ)\ — )^(\\uNlf2+s + \\uN2f2+s)
< c{E\Tm{nx)f^)^rs.

266 Chapter 9 Strong Approximations
By the Holder inequality and Lemma 1.2.3, we have
Wi-l oo Νχ-1
/3 = Я|Гт(щ)|§{ Σ XjJ2XN1+k- Σ u"2Xi
j=N2+l fc=l j=N2+l
Ni-1 oo Ni-1
-Ε Σ ΧίΣΧΝ!+Ι<: + Ε Σ uN2Xj}
j=N2+l fc=l j=N2+l
<с(Е\Тт(щ)^)Ш
3 к
2
2+6
<α(Ε\Τη(ηι)\Ψ)πι
EE^Ma(Nllk-j)hNi+k-j)2l6y
<с(Е\Тт(П1)\2-?)Ш,
where /(χ(2+δ>/2) = g(x), inv/(x) - (inv#(a;))(2+6)/2. For /4, we have
/4 = £?|τ,Λ(η1)|ΐ(ΣΣ+ΣΣ)
j=l1<г<7 г=1 1<7<г
* (^+#2-Л+.7+#2-1 "" ^^i+iV2-l^+j+iV2-lJ
П2 712 ι
j=l1<г<7 t=l 1<J<г ^ >
■(Е\Тт(П1)\2-?)Ш,
where f(x2) = #(#), inv/(#) — (inv^(x))2. It is easy to see that
712 712
j=l l<i<j г=1 1<7<г
1 / 1 \2 2+6 δ
■a(i)z+*mvg(—r-) (Е\Тт(щ)\ 2 )2+s
<c(E\Tm(ni)\*¥)£*.
Thus combining these with (9.3.4) and (9.3.5) we have
тп<тП1+М8тП2+ст8п№+8\
Hence condition (c) of Proposition 9.1.1 is satisfied and the proof of
Theorem 9.3.1 is completed.

9.3 Strong approximations for a α-mixing sequence
267
Corollary 9.3.1. Let {Xn,n > 1} be an α-mixing sequence, g(x) =
xr, r > 2 + δ, Ο < δ < 2. If
(г) σΐ = Ε Si > Cn, for some С > 0,
(**) En=iOc(n)^s~r <oo,
then
Sn - W{al) = 0(ai* (log ση)1+(1+λ)/(2+δ)) a.s.
Particularly, if δ — 2 and limna2/n — σ2 (without loss of generality,
assume that σ2 — 1), we have
Sn - W(n) = O^/^logn)3/2) a.s.
Furthermore, if supn \Xn\ < oo, Σ^η)1^ < oo and σ2 — ησ2, then for
any ε > 0
5n - W(n) = Oin^^logn)5/4^). a.s.
Shao (1989a) gave the following theorem and corollary, which improve
the results in Bradley (1983) and Dehling (1983) under the weaker
conditions.
Theorem 9.3.2. Let {Xn,n > 1} be an α-mixing sequence with
EXn — 0, supn Ε\Χη\2+δ < oo for some δ > 0. Assume that
sup sup E\Sk(n)\2+6/n1+*/2 <oo, (9.3.6)
п>1к>0
a(n) = (9((logn)-r), r > 1 + 2/δ. (9.3.7)
TTien for any 0 < θ < \(£г$ — 1)5 we have
S(t) - W(a2) = 0{tll2{\ogt)-e) a.s.
Corollary 9.3.2. Let {Xn,n > 1} be an α-mixing sequence with
EXn = 0, supn£;|Xn|2+<5 < oo(<5 > 0). Assume that a(n) = (9(n~r)
/or r > 1 + 2/δ and
lim σ2/η = σ2 > 0.
Then {Xn} Obeys the law of the iterated logarithm and the Chung law of
the iterated logarithm.
The proofs of Theorem 9.3.2 and Corollary 9.3.2 are omitted.

Chapter 10 The Increments of Partial Sums
In Chapter 9, we studied strong approximations of partial sums of a
mixing sequence by a Wiener process. But, with the help of these theorems,
it is not enough to obtain the ideal increment results, similar to that in
the case of i.i.d. sequence (see M.Csorgo and P.Revesz 1981). In this
chapter, we intend to study the increments of partial sums of a sequence
of (^-mixing random variables, which may be non-stationary, even non-
identically distributed, by a direct approach. Under the restriction on the
rate of convergence to zero for mixing coefficients, the following results
are close to those corresponding to independent random variables. Our
method can be used to deal with other kinds of mixing sequences as well.
10.1 Some lemmas
In order to show increment results, we need to establish some
exponential inequalities. To this end, we first give the next result (cf. Stout
1974, Lemma 5.4.1 and its corollary).
Lemma 10.1.1. Let {Zn,Tn,n > 1} be a supermartingale with EZn —
0. Let Zo — 0 and U% = Z{ — Z;_i for г > 1. Suppose Щ < С a.s. for some
0 < С < oo and all г > 1. Fix λ > 0 such that XC < 1. Put
η
Mn = exp(AZn)exp{-(A2/2)(l + AC/2) ^^l^-i)}
г=1
for η > 1 and Mq = 1 a.s. Then {Mn,Tn,n > 0} is a nonnegative
supermartingale, and further,
P{supMn > a\ < a-1
for any a > 0.
Let {Yn} be a sequence of (^-mixing random variables with mixing
coefficients φη = φ{η) [ 0. Without loss of generality, assume that EYn —

270
Chapter 10 The Increments of Partial Sums
η
0 for every n. Put Tn = ^ Y^ τ2 — £?T^. In this section, we always
t=l
assume that the following conditions are satisfied:
(a) \Yn\ < bn < oo;
(b) there exist 0 < σ2 < σ'2 < oo, such that σ2η < £(Ут+1 Н h
^n+n)2 < σ1 η for every га > 0 and all large n.
Let p,q,k be integers satisfying that ρ = pn < n,q = qn = o(pn),
gn t oo, /с = A:n = [n/(pn + qn)]. And put Ь = max b{.
l<i<n
Lemma 10.1.2. Suppose that
<pqpb2 = o(l), Σ V1/20p) = ^ i10·1·1)
TTien /or ж = xn and small ε > 0 satisfying
-bn<j>q<x< Щ- (10.1.2)
ε pb
and
x2/n -► oo, (10.1.3)
P{ max |Γί| > ,} < 3 ехр{-Ц|^} (10.1.4)
/or all large n. If (10.1.2) is replaced by
*>^£, (10.1.5)
then
P{iWJTi\ > x} < 3 exp{-^% (Ю.1.6)
Proof. We always assume that η is large enough. First, we prove
(10.1.4). Define
(i+l)p+iq
fc = Σ y;>
j=i(p+q)+l
(i+l)(p+q)
Vi= Σ lj-, г = 0,1,···,*: — 1,
j=(i+l)p+iq+l
j=k(p+q)+l

10.1 Some lemmas
271
Put σ-fields JF_X = {0, Ω},^ = a(Yj, j < (i + l)p+iq), г = 0,1, · · · ,k- 1.
Define martingale differences л — ξί — £J(^| J^-i), г = 0,1, · · ·, к — 1. Write
ρ{\τη\ >χ}< p{i Σ&Ι > (ι - \)χ) + p{\ Σ*·ι ^ Μ
=: Λ + J2. (10.1.7)
Consider </χ. Write
Jx < Ρ{\Σ,-Κ\ * (1-ε)χ} + Ρ{\ΣΕ(.ϊ№-ι)\ > \ε*}
i=0 г=0
=: Jn + J12. (Ю.1.8)
By Lemma 2.2.8 (note |&| < pb), for any Д-χ G .T^-i,
|^^/Bi_1|<2^pbP(Si_i) (i = 0,l,...,fe-l),
which implies
|S(6l^i-i)|<2^i* a.s. (i = 0,l,...,fe-l). (10.1.9)
Using condition (10.1.2), we obtain
J12 - 0. (10.1.10)
Next, we estimate Jn. Since {7^^·,г — 0,1, ···,& — 1} possesses
martingale difference property, by using Lemma 10.1.1 and noting |7;| <
(1 + e)pb a.s. for large n, for 0 < λ < ((1 + e)pb)~1
0=ехр(А^7г)ехр{-у(1 + ^(1 + £)рЬА)^^|^-1)}
г=0 г=0
(j = 0,l,---,fc-l)
possess nonnegative supermartingale property. Write
fc-i
i=o
Ρ{Σ7ί>(1-φ}
= P{&-i > exp(A(l - е)ж)
• exp{-y (l + 1(1 + e)pb\) f \Ε{Ί№-ύ}}- (Ю.1.11)
г'=0

272 Chapter 10 The Increments of Partial Sums
We are going to estimate Σί=ο Ε(^\Τί-ι). Write
fc-i fc-i k-i
Σ Β(7?|*-ι) = Σ S(f?l*-i) - Σ(Β(&Ι*-ι))2
г=0 г=0 г=0
We have |J5(&2|^_i)-J5&2| < 2<pq{pb)2 a.s. by a similar proof to (10.1.9).
Hence, by conditions (b) and (10.1.1), we get
Σ£?(62|^-ι) = (1 + ο(1))Σ^2 a.s.
г=0 г=0
Moreover inequality (10.1.9) implies that
fc-i
Σ(β(6|^-ι))2 < 4<^b2n - o(n) a.s.
Thus
fc-l fc-l
Σ^(7ζ2|^-ι) = (1 + ο(1))Σ^·2 a.s. (10.1.12)
г=0 г*=0
fc-1
Now we estimate J] Εξ2. Write
i=0
fc-l fc-l
Σ^2 = ^(Σ^) -2 Σ *fcfc> (Ю.1.13)
г=0 г=0 0<t<7<fc-l
where |£7&0Ι — 2^((j — г — l)p)pba'p1l2 by Lemma 1.2.10 and so for fixed
г
oo
Σ |i%&| < 2σ'^ν/2?>Σ^1/20» = «(Ρ)
j>i+2 j=l
by assumptions in (10.1.1). Summing on г leads to o{n). Also
|^i&+i|<2^iA^P1/2 = o(p)
by (10.1.1) and summing leads to o(n) again. Thus
Σ£# = s(5>)2 + o(n). (Ю.1.14)
Furthermore
k-
Ε(Έ^Υ = ETn - 2ETn(j2Vi)+E(J2viY. (Ю.1.15)
fc_1 ,2 0 , * N , * N2
г'=0 г'=0 г=0

10.1 Some lemmas 273
By condition (b) and the second equality in (10.1.1), we can prove that
к 2
β(Σ«) =0(kq + p).
In fact
к k—1
Efevif <2Я(^>)2 + 2Ет£,
i=0 t=0
where Εη% = Ο (ρ) and
Ε(Ση)2 = ΣΕτα + 2 Σ E™j
i=0 г=0 0<i<jf<fc-l
k-1
< a'2kq + 2a'2q^(k - J>1/2(jp) - O(fcfl).
i=i
Hence (10.1.15) implies that Ε(ΣΪ=ο &)2 = U + °(1))rn- Combining it
with (10.1.14) and (10.1.12) implies that
fc-i
ЕЯ(7?|^-1) = (1 + о(1))т^ a.s. (10.1.16)
Inserting it into (10.1.11) and using Lemma 10.1.1, we obtain
fc-i
ρ{5>>(ΐ-Φ}
< exp{—λ(1 — ε)#}
•exp{y(l + i(l + £)pbA)(l + e)rn2} (10.1.17)
for all large n. Choosing λ = ж/((1 + ε)Τη), we have λ < ε((1 + е)рЪ)~1 by
(10.1.2). Inserting it into (10.1.17) implies that
ρ{Σα > (i - Φ} < ехР{-(1~Лф2}· (ю.1.18)
Replacing Y^· with — Yj, we obtain
Jn - p{i I>i ^ (1 -£H ^2 exp{~(1 ~ο42)χ2}·
г=0 ™

274
Chapter 10 The Increments of Partial Sums
Combining it with (10.1.10), we have
(1 - 4ε)χ2
,l£2exp{_iiz^}
2ri
For J25 it is clear that 3-1 exp{—(1 — 4ε)χ2/(2τ%)} is one of its upper
bounds in view of qn — o(pn). Then we have proved
P{|Tn| >x}> 1ехр{-Ц^}. (10.1.19)
In order to obtain (10.1.4) from (10.1.19), we use Lemma 5.1.2. First of
all, since φη [ 0, there exists an interger no such that φηο < 10"1. Using
condition (b) and (10.1.3), we obtain
max Р\\ТП-ТЛ > .EX* \
i<i<n V г| ~ 2(1 + ε)1
< max
1<г<п
(! + *)■
4(1 + ε)2Ε(Τη - Ti)2
\2^Κ
4(1 + ε)2σ"η 1
ε2χ2 ~ 10
for large η. So 77 in Lemma 5.1.2 can be chosen as 5-1. Furthermore
/2 /2,
,,„, , εσ η εσ κ . . .
max \YA <b< < —^-x = o(x) (10.1.20)
l<i<n px X2
by (10.1.2) and (10.1.3). Hence
P{ max \Yi\ > εχ/(2(1 + e)(n0 - 1))) = 0
for large n. Therefore, from the above results and Lemma 5.1.2, we obtain
P{ max \TA > x)
l 1<г<п J
< -p\\Tn\ > -^-\ + -P\ max |У<| > Щ -}
~ 4 V ' ~ 1+εί 4 li<t<n' г| ~ 2(l + e)(n0-l)J
<3exp{-^|^x2}.
This is namely (10.1.4).
Now we prove (10.1.6). It is clear that we can assume x/pb > δ for
some δ > 0. If condition (10.1.5) is satisfied, choosing λ = ε((1 + ε)ρύ)~λ
in (10.1.17), we have
P{5> > (1 - Φ) < ехр{-£(12"^Ф}· (Ю.1.21)

10.1 Some lemmas
275
By imitating the procedure from (10.1.18) to (10.1.4) and replacing (10.1.20)
by max |Yi| < b <
lemma is proved.
by max \У{\ <b< χ/{ρδ) = o(x), we obtain (10.1.16) from (10.1.21). The
1<г<п
Lemma 10.1.3. Suppose that condition (10.1.1) in Lemma 10.1.2 is
replaced by
к
¥>рдЬ2 = о(1), Σν1/2(ίΡ) = 0(1). (Ю.1.Г)
i=i
Then for any ν > 0, there exists an α(ι/), such that for all large η and a
satisfying a > a(i/), we have
P{Tn > arn} > ^exp{-(l + v)o?/2}
provided that
φς = o(a/b) (10.1.22)
and
рЪатп = o(ra). (10.1.23)
Proof. For 0 < δ < 1 write
k-i к
P{Tn > ατη} > Ρ{Σ & > (1 + 5)arn} - P{^> < -δατη}
i=0 i=0
=: /i - /2 (10.1.24)
and
k-l k-l
/ι > Ρ{Σ ^ > (1 + 2^)arn} - Ρ{Σ β(&№-ι) < -<5ατη}
г=0 г=0
=: hi ~ /i2. (10.1.25)
Using condition (10.1.22) and recalling (10.1.9), we also have for large η
Iu = 0. (10.1.26)
We are going to estimate 1ц. Let ε be a small positive number indicated
later. Put ν = 3>/e/(l - Зу^),^ = (1 + 2δ)α/(1 - ν). Write
fc-i
#{ехр[и(]П 7г)/тп] | Fk-2}
t=0
fc-2
= exp{ix(5^7i)/rn}£;{exp(iX7fc-i/rn)|^b-2}. (Ю.1.27)
г=0

276
Chapter 10 The Increments of Partial Sums
Condition (10.1.23) implies
ирЪ/тп -> О. (10.1.28)
Hence
£{exp(u7fc_i/rn)| Tk-2)
Inserting it into (10.1.27) we have
mm4|Vm-^-^)
fc-2
•^(7|-1|^-2))]|^-2}>ехр{^(^7г)/гп}.
Thus , by induction,
^{expKg7i)/rn]exp[-^(l- ψ) |>(7^-i)]} > 1·
Since we also have (10.1.16), the above inequality can be rewritten as
*Μ«(Σ ■*)/*]} ^τΟ-^-ϊ)}
>exp{f(l-|)} (Ю.1.29)
for given ε > 0, provided that η is large enough.
Then the following proof is similar to the corresponding оде of Theorem
5.2.2 in Stout (1974) except that some details are different (noting the
choice of υ and и ). It is omitted.
In addition to Lemmas 10.1.2 and 10.1.3, the following generalization
(see Erdos 1959) of the Borel-Cantelli lemma will also be utilized for the
proof of our theorem.

10.2 How big are the increments when the moment generation functions exist? 277
Lemma 10.1.4. If Ci,C2,· · · are arbitrary events satisfying the
conditions
oo
Σ P(Cn) = oo
n=l
H£Sf Σ Σ PiPuCi)l(£ P{ck)f = ι,
k=l 1=1 fc=l
then
P{Cn,i.o.} = 1.
10.2 How big are the increments when the
moment generation functions exist?
Let {Xn} be a sequence of (^-mixing random variables with EXn —
0 (n > 1). Put Sn = Σ Xfc, afc2 - £X2.
fc=l
Theorem 10.2.1. (Lin 1991) Suppose that {Xn} defined above satisfies
the following conditions:
ft) l™n-+oo infm>0 E(Xm+1 Η \- Xm+n)2/n > 0;
(ii) there exist to, Μ > 0, зг/c/i that EetXk < Μ for every к and any
\t\ < to]
(Hi) there exists an I > 1, such that φη — 0(n~l).
Suppose that {an} is a non-decreasing sequence of positive integers
satisfying
(a) there exists an a > 0 such that α log η < an < η where d >
(31 + 1)/(/ - 1). Then, putting
σηΝ — Ε(Χη+ι Η l· Χη+άΝ) ,
βηΝ = anN{2[\og(N/a2nN) + log log TV]}1/2,
we have
Umsnp fiJ^N\S(N,aN)\ = 1 a.s. (10.2.1)
N-+oo
limsup max /Зп^|5(?г,адг)| = 1 a.s. (10.2.2)
lim sup max 0^N\S(N,k)\ = l a.s. (10.2.3)
JV-+00 l<k<aN

278
Chapter 10 The Increments of Partial Sums
limsup max max β N\S(n,k)\ = 1 a.s. (10.2.4)
N—+oo 1<τι<Ν 1<&5:αΝ
Furthermore, if condition (a) is replaced by
(b) there exists an a > 0, such that an1'^1' < an <n and
lim log(ri/an)/log log η = oo,
п—куэ
then
lim max■ β-^\Ξ(η,αΝ)\ = 1 a.s. (10.2.5)
N—юо l<n<N
lim max max /3~^|5(n,fc)| = 1 a.s. (10.2.6)
N—юо l<n<N 1<к<ан
Proof. From conditions (i)—(iii), there exist constants w\ and W2,
0 < w\ < W2 < oo such that
wxn < £(Xm+i + · · · + Xm+n)2 < w2n (10.2.7)
for every integer m > 0 and η large enough. The right inequality sign is
due to that from Lemma 1.2.8
\EXiXj\ < 2^^£;1^|X.|^jE;i-^|Xi|^/(^-a)
= ο(ϋ-ί)"ι/ι#)
for 1 < I' < /, г < j.
First, we prove (10.2.1) —(10.2.4). Obviously, it is sufficient to verify
limsupβ]ν1Ν\3(Ν,αΝ)\ > 1 a.s. (10.2.8)
N—►00
limsup max max /3~^|5(η,k)\ < 1 a.s. (10.2.9)
N—юо 1<η5··^ ΐ5··^5ίαΝ
For Β > 1/ί0, define
71
Yn = XnI(\Xn\<Blogn), Yn' = Yn-EYn, Tn = £n',
fc=l
ληΝ = ^(^n+l + * * * + Yn+aN) > ^(^j fc) = 7n+fc — ?n,
otnN = λη^ν{2[1ο§(ΛΓ/λ^) + loglogiV]}1/2.
From condition (ii), P{Xn Φ Yn, i-o.} = 0 and for t' such that t' < to and
t'B> 1
\EYn\ < сп~1'в,
max max /3-^|S(n,fc)-T(n, fc)|->() a.s. (10.2.10)
Hence, as TV —> oo
max η
L<n<7Vl<fc<a;v

10.2 How big are the increments when the moment generation functions exist? 279
Moreover, it is not difficult to show by (10.2.7) and the definition of Yn
that
<&v/<&v^l (N^oo) (10.2.11)
uniformly in n.
We are now going to prove (10.2.8), which is equivalent to
limsupa^|T(AT,a^)| > 1 a.s. (10.2.12)
N—*oo
Put h — 2/(3/ + 1). Define pn = [nh], qn = [anpn], where {an} is a
sequence of positive numbers tending to zero slowly enough. By conditions
(iii), (a) and the definitions of pn and qn , it can be seen that
VqaNPaN(logNf < ca-lNp-lN+1(\ogN)2
< ca-lNa-N2il-1)/{3l+1\\ogN)2 ^ 0, as JV -+ oo
provided that an tend to zero slowly enough. We also have
Σ<Ρ1/20Ρ) < cJZiJPr1'2 < ckn-1'2 < m(3*-i)/(3*+i)-*/2 = o(1)
with the help of condition (iii). Hence condition (10.1.1), and further
condition (10.1.1'), are satisfied. Similarly, we have
φ4αΝ = o((\og(N/X%N) + loglogiV^/log JV)
and paN (logN)anj\f/'apt = o(l). Hence, we can use Lemma 10.1.3 (choosing
ν = ε > 0), which implies that for large TV
P(CN) > i exp{-(l + e)(i - e)2log(JV/A^) + log log N}}
where С ν = {ol~^nT{N, αχ) > 1 — ε}. Put ηχ = 1 and. define n^+i =
n^ + 2anfc. Noting that the sequence is mixing, we have
771 771 771 ~
|ΣΣρ(α*σο/(Σρ(σο) -ι
<
ЕГ=1 P(cn,.)(l - P(Cn.)) + 4Σ^ι Ebj+i P(<?n>(nfc - nj ~ %·)
(Er=i^(Cnj)2

280 Chapter 10 The Increments of Partial Sums
m m
< (l + 4 Σ <p(nk - щ - αηι))/Σ P(Cnk).
k=2 k=l
Using condition (iii) and recalling the definition of n^, one can verify that
oo
Σ ф(пк — п\— аП1) < oo. Moreover Σ5£ι P{Cnk) — oo since
k=2
]T (nlogn)"1 < (nfclognfc)_1(nfc+i -nk)
7l=7lfc + l
= (nklognk)~12ank < cP(Cnk).
Hence, from Lemma 10.1.4, P{Cnk,i.o.} = 1, which yields (10.2.12). The
proof of (10.2.8) is completed.
Furthermore, we want to prove (10.2.9). From (10.2.10) and (10.2.11),
it is sufficient to show that
limsup max max апм\Т{п,к)\ < 1 a.s. (10.2.14)
N—юо l<n<Af 1<^<α;ν
Let r — r(e) > 1 be a positive integer indicated later. Put R — [адг/г], nr —
R[n/R). We have
\blN/E(Ynr+1 + · · · + Y{n+aN)rf - 1| < 2Wf^'^ (Ю.2.15)
for large N by (10.2.7) and (10.2.11). Write
\Tn+k - Tn\ < \Tn+k - T(n+fc)r| + \T(n+k)r - ТПг I + \ТПг -Тп\. (10.2.16)
Consider the second term of the right hand side of (10.2.16). We have
Pi LS i§gw <N\T{n+k)r ~ Tnr I > 1 + ε/3}
< cr— max Pi max \T(„ , j.\
~ aN i<n<N h<k<aN ' ^n+K>r
- Tnr\ > (1 + εβ)αηΝ}. (10.2.17)
Obviously, |(n + aff)r — nT\ < адг(1 + 1/r). We can use (10.1.4) of Lemma
10.1.2, since
(log Ν)αΝφ4αΝ =ο(αηΝ),
anN = o(E(T(n+aN)r - Tnrf/paN logiV).

10.2 How big are the increments when the moment generation functions exist? 281
Noting (10.2.15) and choosing r to be large enough, we know the right
hand side of (10.2.17) does not exceed
cr^exP{-(l-e/3)(l + e/3)2a2nN/E(T(n+aN)r-Tnr)2}
N / aN \ i+e/4
-^^VJVlogTV/
= Cr(M)^(logiV)-(1+e/4)
for large N. Let N\ = I and define Nj+i by a^j+1 = min{an : an >
[ej]}(e > 1). Obviously, Nj+1 > [0i] since aN < N. Thus
which implies
lim sup max max a~^ |T(n+fc)r - T„r | < 1 + - a.s. (10.2.18)
From conditions (i)-(iii) and (10.2,. 11) and the definition of Nj, it is easy
to see that there exists a constant G > 0, such that
lim
a;
2
■nNj+i
urn Л2
<G{e-l)1'2. · (10.2.19)
Now, choosing θ near enough to 1 and noting that the ranges in the two
max's in (10.2.18) enlarge as j increases we have from (10.2.18)
limsup max max а~^\Т(п+к)г -ТПг\ < 1 + - a.s. (10.2.20)
N—юо 1<η<Ν 1<к<ам ^
We turn to the first term in the right hand side of (10.2.16). Write
P< max max OL~\r\Tn+ j, — T<n , u\ I > - \
\l<n<Nl<k<aN ηΛΠ + (n+/e)rl - g/
rN n ( lrri
< max max P\ max \Tj
адг n к l(n+k)r<j<(n+k)r+R
-T(n+k)r\>^anN}. (10.2.21)
Using a similar proof to (10.2.18), one can also employ (10.1.4) of Lemma
10.1.2. By recalling (10.2.7) and (10.2.11) and choosing r = r{e) to be

282
Chapter 10 The Increments of Partial Sums
large enough, the right hand side of (10.2.21) does not exceed
aN I 36£(T(n+fc)r+fl -T(n+fc)r)2 λ^
όΠ\ ( 2l IV lOg IV ϊ
< exp< —ere log >
<3rN, aN .cr* 2
- ал/ \NlosN/ ~ &
for all large N. Thus
lim sup max max a~l \Tn+k - T(n+k)r \ < - a.s.
By imitating the procedure from (10.2.18) to (10.2.20), we have
limsup max max a~^|Tn+fc - T{n+k)r \ < - a.s. (10.2.22)
7V_»oo l<n<N 1<к<ам 4
Obviously, we also have
-1 ι £
limsup max anN\Tn - ТПг\ < - a.s.
7V_>oo l<n<N 4
Therefore, (10.2.14), and further (10.2.9), are proved.
Finally, we prove (10.2.5) and (10.2.6). For this purpose, from (10.2.2)
and (10.2.4) proved, it is sufficient to verify that
liminf max /37^|S(n,ajv)| > 1 a.s. (10.2.23)
N—ЮО 1<TI<N
By (10.2.10) and (10.2.11), (10.2.23) is equivalent to
liminf max a~^\T(n,aN)\ > 1 a.s. (10.2.24)
Ν—>οο 1<η<ΛΓ
We have for large N
P{11^NanN\T(n,aN)\ < 1 -ε}
- pLi<[^]-^^-Ar|r(2jaAr'ajv)l -λ -ε}
^ Π P{a^aN>N\T(2jaN,aN)\ < 1 - ε}
N \l-e/2
^адг/

10.3 How big are the increments when the moment generating functions do not exist?283
(by (10.2.13) and condition (b))
1-εΊ [Ν/αΝ]1-^2
+ C[
<c(logiV)-2.
< 2exp{-c(^)-£/2(logAr)-(^)} + c(^\
The last inequality is due to condition (b). Thus, if Nj is defined as above,
we have
liminf max a~l \T(n,aN.)\ > 1 a.s. (10.2.25)
Considering Nj < N < Nj+χ, we get
liminf max а~1г\Т(п.а^)\
N-+oo 1<η<Ν ηΛΜ K ' Л
> liminf max (α^Ι\Τ(η,αΝ.)\)(αηΝία^)
-limsup max max а^\Т(п,к)\.
TV—юо 1<η<^ν 1<к<ан—ан.
The first term in the right hand side is a.s. > 1 — G'{9 — l)1/2 for some
G' > 0 by (10.2.25) and (10.2.19). The latter is a.s.< G"{1 - 1/0)1/2 for
some G" > 0 by (10.2.14). Letting θ [ 1, we obtain (10.2.24). The theorem
is proved.
Remark 10.2.1. When / in condition (iii) is large enough, in other
words, φη tends to zero at a great rate, an can be 0(log3+£ n), for any given
ε > 0. For an independent sequence, it is required that an/logn -^ooas
η —> oo (see Lin 1988).
Remark 10.2.2. In the theorem, we don't require that an/n is non-
increasing. But it is assumed even if a sequence is independent (cf. Csorgo
and Revesz 1981). In fact, this condition is not realistic, since either an = η
for all η or an = m Д η for some fixed m when the condition is added.
10.3 How big are the increments when the
moment generating functions do not exist?
Let {Xn} be a (^-mixing sequence mentioned in the beginning of the
above section.

284
Chapter 10 The Increments of Partial Sums
Theorem 10.3.1.(Lin 1989) Suppose that {Xn} satisfies condition (i)
in Theorem 10.2.1 and
(ii/ there exists a non-decreasing continuous function H{x),x > 0,
such that
oo
J2 P{H{\Xn\) > Μ < oo for any δ >0, (10.3.1)
n=X
Km" \ Σ E(H(\Xj\)^ < Μ < oo (10.3.2)
j=n+l
uniformly in η for any β < 1,
χ~(2+Ί'Η(χ) is non-decreasing for some 7 > 0, (10.3.3)
lim H(x/2)/H(x) > 0, (10.3.4)
χ—>οο
(Hi/ there exists an I > 1 + - such that φη — 0(n~l).
Let {an} be a non-decreasing sequence of positive integers satisfying
(a/ there exists such an a > 0 that a(invH(n))d < an < η where
d = 2(i + l)/(l - 1).
Then the conclusions of Theorem 10.2.1 remain true.
Proof. The proof of the theorem is similar to that of Theorem
10.2.1. We outline the main differences. It is easy to verify that condition
(10.3.1) can be replaced by
00
Σ Ρ{Η(\Χη\) > δηη} < oo (10.3.5)
n=X
for some δη | 0. Define Yn = X„/(|X„| < inv#(6nn))X = Yn-EYn,Tn =
Σ Yi By (10.3.5) we have
k=l
P{XnyiYn,i.o.} = 0. (10.3.6)
Without loss of generality we can choose δη such that η~ε = ο(δη) for any
ε > 0. Prom this and using the conditions EXj = 0, (10.3.2) and (10.3.3),

10.3 How big are the increments when the moment generating functions do not exist?285
we have for all large n,
n+fc n+fc «
j=n+l j=n+lJ{Hy\Xi\>>Si3i
<c Σ (6jj)-^E(H(\Xj\))^!+^
j=n+l «
ϊίί 3+7 72+57+8
<c ς г**:ε(η(\χΑ))ί'2+6ί+8
j=n+l
n+k
7^+5.57+8 . JL
. 9 ι *. . ο Ι -V^
<cfc~6+7J J2 E(H(\Xj\)) 72+67+8 J
л 7^+5.57+8
^ννκ^ι^ ' /
.7=71+1
<c№. (10.3.7)
Consequently (10.2.8) is also equivalent to (10.2.12). Put h = l/(/ +
1)? Pn = [™Λ]>9η = [<*nPn], where an are positive numbers tending to zero
slowly enough. Using conditions (iii)' and (a)', we obtain
VqaNPaN(invH(6NN))2
<ca-lNa-Jlil-1\mvH(6NN))2
< ca-4mvH(6NN)/mvH(N))2.
By condition (10.3.4) one can get mvH(6NN)/invH(N) —> 0 (see Lin and
Lu, 1992, (2.3.12)). Hence ,we have
VqaNPaN(H(6NN))2 -► 0, as N -+ oo.
provided that αχ tends to zero slowly enough. Moreover the other
conditions in Lemma 10.1.2 are also satisfied. Then we have (10.2.13) as well.
The following proof is similar to that of Theorom 10.2.1, hence, is omitted.
Remark 10.3.1. When / in condition (iii)' is large enough, i.e., φη
tends to zero at a great rate, an can be 0((ιηνΗ(η))2+ε) for any given ε >
0. For an independent sequence, it is required that an > c(mvH(n))2/logn
(see Lin 1987).

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Chapter 11 Strong Approximations
for Mixing Random Fields
Let {Xj,j G Nd} be a stationary mixing random field with EXj = 0.
The various definitions of mixing have been given in Chapter 6. We shall
also employ the notations .AciC Bd,C$, As, Α(δ), etc, which have been
posed there. We define the partial-sum process by
Sn(A) = Σ \nA n Cjl^j for any Л G Д (11.0.1)
j
where η = (nb · · · ,nd) G Nd,ib4 = {(rii^i, · · · ,ndxd), χ = {χι,-- ,xd) £
A}.
The main subject of strong approximations of set-indexed processes is
to establish a new probability space, on which there exists an independent
identically distributed centered Gaussian field {lj,j G Nd} with covariance
σ2 = EX\ + 2J2j ^-Xj+i-X"i> and without changing its joint distribution,
the field {Xj,j G Nd} can be redefined on this new probability space, such
that
Dn = sup{X;MnCj|(Xj-yj)}
= 0{nd'2-e) or 0{nd'2 (log n)-a) (α,ε>0), (11.0.2)
under some conditions on A and {Xj}.
Denote
|n| = nin2·· "nd, if η = (ηι,···,η^),
G^ = |n= (ni,---,nd), nj > Y[ ra£, i = 1,2, ·· · ,d|, (11.0.3)
fc/г
for 0 < β < 1.
Berkes and Morrow (1981) first discussed the strong approximation for
a weakly stationary α-mixing random field {Xj, j G Nd}. Suppose that

288
Chapter 11 Strong Approximations for Mixing Random Fields
EXX = 0,£|Xi|2+* < oo for some δ > 0 and a{n) = 0(η-^1+ε^ι+2^).
They proved that for any η G G/j and some λ > 0
sup |£(Xi - У0| = Odnl1/2^) a.s. (11.0.4)
This result was improved by Strittmatter (1990), in which the condition
"n G G/з" is left out and the following result is obtained: If
Ss — 4 — 2δ
Ν(ε) := Card(.A(e)) < ce~u, и < —— —- - 2, (11.0.5)
d[4 + ο)
6(e) := sup{n'd\((nA)ne Π (пАс)П£)\ :
Ле (J Αη, η > 1/ε}
η>0
<ceh for some 0 < h < 1, (11.0.6)
a(n) = 0(n~s) for some 5 > 1 + (17/2)d~1/(1 Λ δ), (H.0.7)
then for some 7 > 0
sup J] Μ Π CjK-Xj - 15) = 0{ηάΙ2~Ί) a.s. (11.0.8)
The class A of subsets in the above results which satisfies Ν(ε) < ce~u
is the smaller subset class. Su (1992) proved a strong approximation for
a (^-mixing strictly stationary random field with Η (ε) < οε~τ for some
r > 0, and η G Ωβ. We shall introduce Su's results in Section 11.1 and
Strittmatter's results in Section 11.2.
11.1 Strong approximations of a ^-mixing random field
For simplicity, we only give the results for the case of d = 2.
Theorem 11.1.1. (Su 1992) Let {Xj,j G N2} be a strictly stationary
φ-mixing random field with EXj = 0 and JS|-Xj|2+* < 00 for some δ > 0.
Suppose that the class A of subsets ofB2 satisfies (11.0.6) and the entropy
condition
Η(ε) < αε~τ for some 0 < r < 6/(4(1 + δ)), (11.1.1)
and suppose that
φ(χ) = 0(χ-(1) for some q> c ,* сч "2· (11.1.2)
0 — r[4 + 0)

11.1 Strong approximations of a </?-mixing random field
289
Then
σ2 := EX2 + 2^2 EXiXv = O(l), (11.1.3)
further, without changing the distribution of {Xj, j 6N2}, we can redefine
the field {Xh j G N2} on a richer probability space together with an
independent normal random field {Yj, j G N2} with EYj = 0, ΕΥ? = σ2, such
that
sup £ \nA П Cj|(Xj - Yj) = 0(|n|(log |η|)"σι) ο.*. (11.1.4)
for every η € Gp,6/(2(2 + δ)) < β <l, where σι = a\(r,q,S) > 0.
First, we prove the Bernstein inequality for a y-mixing random field.
Lemma 11.1.1. Let {Xj, j G Nd} be а φ-mixing random field with
EXj = 0, |-Xj| < Δη a.s.l < j < n. Denote ση = maxi<j<n ||-Xj||2· If
Σ™=ι φ1/2(2η) < oo, then for any A £ Bd we have
ρ{|ΣΜης,|χ,|>χη}
< 2exp(c|n|v?jm) - axn + са2\пА\а2Л, (11.1.5)
where xn > 0, m < ηιίηκκ^ί and 2damdAn < 1/2.
Particularly, if a = l/(2d+1mdAn), then
ρ{\Σ\ηΑΠθ№\>χη}
rc\n\<p(m) _ xn c\nA\a* л
~ Pl md 2d+1mdAn ^ (2d+1mdAn)2 У
Proof. Put N = (Nu · · · ,Nd) such that 2m(Ni - 1) < щ < 2mNi,
г = 1, · · ·, d. Denote V = {(г>х, · · ·, υ a) : Vi = 1 or 2, г = 1, · · ·, d}. Define
/v,k = Pi(l),«2(lj] x · · · x [h(d)Md)i ν G V, 1 < к < Ν,
where
ii(i) = {(2А;; + Vi - 3)ra + 1} Л nb
b(i) = {MO + m - 1} Лтг^, г = 1,2, · · · ,d.

290
Chapter 11 Strong Approximations for Mixing Random Fields
Moreover, denote
l<k<N
^v,k = Σ \ηΑ η /v* n cjIxj'
l<k<N
S = 2j ^v, N·
Note that еж < 1 + ж + ж2 if |ж| < 1/2 and 1 + χ < ex for any ж. Therefore
if 2damdAn < 1/2, we have
Eexp(2daAv,k) <exp((2da)2EAlik).
By Lemmas 1.2.10 and 6.2.3 we have
Eexp(2daBv,N)
= Eexp(2da ^ ;4v,k) exp(2dai4v,N)
k^N
< Eexp(2da ^ A^Eexp(2daAViN)
k^N
+ 2φ(τη)\\βχρ(2άαΑ^Ν)\\00Εβχρ(2άα ]T i4v,k)
k/N
(l + 2e1/VM)exp(c(2da)2| пАП IViN\a2n)Eexp(2da £ 4v,k)
k/N
<
k^N
< (1 + 2e1/VM)|N| exp(c(2da)2|ib4 η /ν|σ2) (11.1.6)
where ||/(ж)||оо means the superior norm of f(x). Prom the convexity of
an exponential function and (11.1.6), it follows that
£exp(aS) < i Σ Eexp(2daB^ N)
< (1 + 2e1/V(™))|N| exp(c(2da)2\nA\al). (11.1.7)
Since |N| < (3/2)d\n\/md and (11.1.7) holds also true for £exp(-aS), we
obtain (11.1.5) by the Markov inequality.
Proof of Theorem 11.1.1.

11.1 Strong approximations of a </?-mixing random field
291
First, under the conditions of Theorem 11.1.1, it is easy to see that
(11.1.3) holds true and
lim E( J2 -Xj)2/|n| = cr2 < oo. (11.1.8)
In order to prove (11.1.4) for any η G Ωβ,δ/{2 + δ) < β < 1, we
shall use the truncation, nesting and blocking techniques in the following
lemmas.
1. Truncation Take r small enough (specified later on), let
χ] = ^/(|^|<υ|(1+τ)/(2+δ)),
S'n(A) = Yi\nAnCi\X'y
Lemma 11.1.2. We have
sup \Sn(A) - S'n(A)\ = O(l) a.s. (11.1.9)
Σ E\*i - X'j\ = 0(nd/(2+5)) a.s. (11.1.10)
j<nl
Proof. Since
P{Xj φ Щ = P{\X-3\ > |j|(i+-)/(2+5)} < ciji-d+τ),
we have P{Xj — X'\ φ 0, i.o.} = 0 by the Borel-Cantelli lemma. Thus for
every η G N
sup \Sn(A) - S'n(A)\ < Σ \Xi ~ Xjl < с < oo a.s.
and
A JGnl
Σ вд -*ji < Σ (ij|"(1+t)(1+6)/(2+6)) = o(nd^2^).
i<ni j<ni
2. Nesting For any given η = (nun2) G Gp,6/(2(2 + δ)) < β < 1,
let α, b be least positive integers such that
2a > (loglnl)1^, 2b > |n|^r+5$f), (11.1.11)
where r' > r is specified later on.

292
Chapter 11 Strong Approximations for Mixing Random Fields
Lemma 11.1.3. We have
sup\S'n(A) - S'n(A(2-b))\ = 0(\п\^-Щ аш8ш
A
Proof. For any A G Д, by conditions (11.0.6), (11.1.1) and (11.1.11),
we have
\S'a(A) - S'n(A(2-b))\
<Y\\nAnCi\-\nA(2-b)nCi\
1+T
< c\n\\A Δ A(2~b)\ · |n| w = 0(\n\^L~^})
\Y'\
lAjl
\h(x—£h)\
Lemma 11.1.4. For every η G Gp
sup \S'n(A(2-b)) - S'n(A(2-«))\ = 0(|n|2 (log |n|)^) a.s.
AeA
Proof. Take r, r' and τ' small enough such that
Denote mj = [|n|4<2+*)2 2 ]. Sincen € G^, it is clear that ггц < щАп2,г =
1,2. On the other hand we have
\X! -ЕЩ< 2|п|(1+г)/(2+г), j < η.
Take Δη = 2|η|(1+τ)/(2+δ) in Lemma 11.1.1, we obtain
P{\S'n{A{2-^))) - ^(Л(2-))| > H1/^-*}
<Ρ{\Σ\η(Α(2-(^)\Α(2-*))\(Χ!-ΕΧ!)\ > \\n\^2~iT)
j
+ P{|EK^(2~i)\^(2-(i+1)))|(X] -EX'})\ > \\η\^2-ίτ}
< сехр(-с2{(г'-г)).
(11.1.13)

11.1 Strong approximations of a </?-mixing random field
293
Therefore from (11.1.1) it follows that
Σ P{sup \S'n(A(2-b)) - S'n(A(2-a))\ > \n\^(\og\n\)-^'}
^ Σ Σ Σ p{| W2-(i+1))) - адг-))! > н1^}
neG^ г=а АеА
Ь
<cJ2 ^ехр(Я(2~') + Я(2~('+1))-с2'(Г,_Г)) <ос'
ηζϋβ г=а
which implies Lemma 11.1.4 by the Borel-Cantelli lemma.
Let к = (fc(l),fc(2)) > 0, define tk = (tk(l),tk(2)) as follows:
tk(l) = [exp(fc(l))-/4], tk(2) = [exp(fc(2))-/4], 0 < s < 1/2.
For large η, η G Gp implies tk G Gp if tk + 1 < η < tk+i.
Lemma 11.1.5. For tk G G^, we have
sup max ^(A^)) - S'tк(А(2~а))\
= 0(|tk|1/2|l0g|tk|)-1/4) a.s.
as |k| —> oo.
Proof. Note that if tk < η < tk+1, А С [0, l]2, we have
|гъ4 Δ tkA| < c|n|(ei + ε2),
where
ei = (ik+i(i) -*k(i))M» * = ^2· (11.1.14)
Put mk = [^к|^2+^)|к|~51. If tk G Gp, we have mk < tk(l) Atk(2).
Let Δη = 2|tk|(1+r)/(2+<5) in Lemma 11.1.1, we obtain
P{\S'n(A{2-)) - S'tk(A(2-«))\ > \t^\k(l)'^ + A;(2)-/16)}
<p{|^|(n^(2-a)\tk^(2-a))n
j
• Cj|(X] - ЕЩ)\ > J|tk|*(fc(l)-* + *(2)"*)}
+ P{\^2\(tkA(2~a)\nA(2-a))n
j
• Cj|(X] - EX!)\ > J|tk|*(fc(l)-* + *(2)"ft)}
< cexp(-c(it(l)s/2 + &(2)*/2)). (11.1.15)

294
Chapter 11 Strong Approximations for Mixing Random Fields
For tk £ G73, we have
fc(l)-*/16 + к(2)-»/16 < cOogltkl)"1/4.
It follows that
£ Pisup sup \S'n(A(2-a))
tk€G0 ^eJtk<n<tk+1
-5ik(A(2-a))|>|tk|i(log|tk|)-i}
<c Σ exp(2(log|tkl)^')|tk+1-tk|
tkeG0
• ехр(-с(А;(1)в/2 + fc(2)s/2)) < oo.
Lemma 11.1.5 is proved by the Borel-Cantelli lemma.
Let Ri = [ti(l),tI+1(l)] χ [ί|(2),ίι+ι(2)]. For any A G n[Js>0As, define
A* = \J{Ri : Ri П Α φ 0}, Л» = (J{#l : ^l С А}.
It is clear that i, с i С # and if tk < η < tk+i we have |Л*\Л»| <
c|n|(ei +£2), where E{ are defined by (11.1.14). Similarly, we have
Lemma 11.1.6. For tk G G7?, we have
sup max |£ |(tkA(2-a)\(tkA(2"a)),) η Cj|Xj|
= 0(|tk|1/2(log|tk|)-1/4) o.e. a*|k|-.oo.
3. Blocking To estimate
sup max |£ |(tkA(2-a))* Π C3\X3\
is the key to the remainder of the proof. Let u = (гх(1),гх(2)) > 0, Hu =
{v G N2,tu + ι < ν < tu+1},0 < ρ < /3. Denote
L = {u : Hu С Gp}, Я = (J Яи,
uei.
Au= (J {v€Hu:iu+1(z)
г=1,2
- [exp(u(i)s/4/4)] < «i < iu+i(t)}·

11.1 Strong approximations of a (^-mixing random field
295
For tk G Gp, define t^ = (tf)(l),t^)(2)),p = 1,2, as follows:
*k (m) = δτη,ρ min vm + (l- 6m,p)nm, m = 1,2, (11.1.16)
νι=ηι,\ΐΙφτη
where 5т?р = 1 if m = p; 6miP = 0 if m φ p. Put
/1 = [о,^]и[о,^2)],
h= (J #u\Au,
We have
(t<1)(l),t<!2)(2))<u<k
/3 = U Δ«·
uGL,u< к
|Sl(tk4(2-e)).nCj|(Aj-Yj)
j
J
+ |^l(tkA(2-a)),n/1ncj|yj
j
+ \£\(tkA(2-a)),m2nci\(xi-Yi)
j
+ |^|(tkA(2-a)),n/3nCj|Xj
j
where lj will be specified below.
Lemma 11.1.7. Forth £ @β, we have
sup max IV |(tk^(2"a))* Π 1г П Cj|Xj|
AG^tk<n<tk+ll j
= 0(|tk|*(log |<*|)_σι) a.5. as |k| -> oo
Proof. It is easy to see that
sup max IV |(tk^(2"a))* П h П Cj|Aj
Ae^tk<n<tk+il j
<E|E№n/inCfjlxJ
Kk j

296
Chapter 11 Strong Approximations for Mixing Random Fields
Since 4X)(1) < ctk(2)P, tk2)(2) < ctk(l)P, we have
iiLx)i/iiki<c|ik|-^-^2 and i42)i/iiki < c\tk\-w-^
for £k G Gp. By the Markov inequality and Lemma 6.2.3 it follows that
p{E|E№n/incjl*j| > |tk|x/2(iog|tk|)-^}
Kk j
^ Σ ^{|IZ l^in 7in CjI^jI ^ l*kl1/2(bg |tk|)-^ ikr1}
Kk j
<c^l^n/1|1+5/2/(|tk|1+fi/2((log|tk|)-^|k|-1)2+i)
Kk
< c|tkr^(^)((log|tk|Hk|)2+6. (11.1.17)
Thus Lemma 11.1.7 holds true by the Borel-Cantelli lemma.
Lemma 11.1.8. For tk G Gp, we have
sup max \£ |(^Л(2"а))* П /3 П ОД
= Ofltkl^logltkl)"*1) a.s. as |k| -> oo. (11.1.18)
Proof. As in the proof of Lemma 11.1.7, we have
sup
AeA
max I^Ktk^-^n/anCjIXj
tk<n<tk+1l^
<Σ|Σΐβιη/3π^ι^
Kk j
Denote Ди = Ди(1) |JAU(2), where Au(l) and Ди(2) are disjoint
rectangles, such that
|Ди(1)| =[ехрН1Г/4/4)]([ехр((и(2) + i)-/*)]
-[ехр(К2)Г/4)]),
|Δ»(2)| =[ехр(п(2)^4/4)]([ехр((«(1) + Ι)*'4)]
-[ехр(Н1)Г/4)]).

11.1 Strong approximations of a </?-mixing random field
297
We have
£ |Ди(1)| < ckilY-'HbWHbp),
ueL
u<k
Σ |AU(2)| < ck(2)1-/4tk(2)1/4tk(l).
uGL
u<k
Then, similarly to (11.1.17) in Lemma 11.1.7 we have
p{E|Ei^n/3nCji^j|^itkil(bg|tkir<Ti}
Kk j
< cj; № η /si^^/iitk^+^ciiog itbD-^ikr1)2^)
Kk
<c\tk\-^^^\(log\tk\r\k\)2+e
■ (k(l)1-*/4 + k(2)1-s/4)1+S/2. (11.1.19)
Lemma 11.1.8 holds true by the Borel-Cantelli lemma .
4. Constructions of {Xj} and {lj}.
In order to construct random fields {Xj} and {lj}, we quote the
following results without any proofs. Denote
hu = Card((#u\Au) Π Ν2), Xu = Σ ХЖ/2·
j(E#u\Au
Proposition 11.1.1. (Berkes, Morrow 1981) Under the conditions
of Theorem 11.1.1, there exists a £, 0 < t < 1 such that for any |ж| < h1^
we have
\Eexp(ixXu) - exp(-aV)| < ch~K (11.1.20)
Proposition 11.1.2. (Berkes, Philipp 1979) Let {Xk,k > 1} be
a sequence of random variables with values in i?dfc, and let {Fk^k > 1}
be a non-decreasing sequence of σ-fields such that Xk is .^-measurable .
Finally, let {Gk,k > 1} be a sequence of probability distributions on Rdk
with characteristic functions #fc(u),u £ Rdk, respectively. Suppose that
for some non-negative numbers \к,$к and Tk > 108dk
E\E{exp(i(u,Xk))\Tk} - 9k(n)\ < Xk (11.1.21)

298
Chapter 11 Strong Approximations for Mixing Random Fields
for all u with |u| < Tk and
Gk{u : |u| > l-Tk] < 6k. (11.1.22)
Then without changing its distribution we can redefine the sequence {Xk,
к > 1} on a richer probability space together with a sequence {Υ&, к > 1}
of independent random variables such that Yk has distribution Gk and
P{\Xk ~Yk\>ak}<ak к = 1,2, · · · (11.1.23)
where αχ = 1 and
afc = Ш^1 logTfc + 4λ*/2Τ^ + 6k. (11.1.24)
Proposition 11.1.3. (Berkes, Philipp 1979) Let 5г,г = 1,2,3 be
separable Banach spaces. Let F be a distribution on 5χ χ 52 and let G be
a distribution on 52 x 5з such that the second marginal of F equals to the
first marginal of G. Then there exist a probability space and three random
variables Ζ^,ΐ = 1,2,3 defined on it such that the joint distribution of Z\
and Z<i is F and the joint distribution of Z<i and Z3 is G.
Let (ц '(1),40 (2)) be large enough such that for any u G Lq := {u G
b,u > (4^(1),4о}(2))}' there exists a P' > ° satisfying
tu+i(l)-iu(l)-[exp(u(l)-/4/4)]
> (iu+i(2) " *u(2) - [exp(u(2)5/4/4)])P\
^u+l(2)^u(2)-[exp(u(2)5/4/4)]
> (Wi(l) " ^u(l) " [exp(u(ir/4/4)])p/.
Put φ is a one to one correspondence from {1,2, · · ·} to Lq, denote φ{1) =
{φι(/), V^C))· By Proposition 11.1.1, there exists a i, 0 < ί < 1 such that
for any |ж| < h1,,* we have
\Eexp{ixXm) - exp(-aV/2)| < сЛ"^. (11.1.25)
On the other hand, since h^^ = 0(|^(j)|), for any \x\ < h1,^ we have
Xi(x) := Ε\Ε{βχρ(ίχΧφ{ι))\Χφ(1),- · · ,Хф(1_г)} -exp(aV/2)|
< Ε\Ε{βχρ(ίχΧψ(1))\Χψ(1),···,Χψ(1_1)} - Еехр(гхХф(1))\
+ \Εβχρ(ίχΧψ(ΐ)) ~ ехр(а2ж2/2)|
< ch-^ + 2π^([βχρ(^ι(/)β/4/4)] Λ [exp(V>2(/)s/4/4)])
<c\4(l)\~t0

11.1 Strong approximations of a ^-mixing random field
299
by the property of (^-mixing, where to = min(£, pq/8).
Put t' < to/2, Τι = Цт\*\ λι = |ίψ(0|-*°, δι = Ν(0,σ2){χ : \χ\ >
Τι/4}, щ = 16Tz_1logTz + 4А*/27) + 6h By Proposition 11.1.2, without
changing its distribution we can redefine the sequence {Χψ^} on a richer
probability space together with a sequence {Υψ(ΐ)} of independent normal
random variables such that ΕΥψη\ = σ2 and
P{\Xm-Ym}\>ai}<ai.
(11.1.26)
Moreover, by Proposition 11.1.3, there exist two random fields {Xj, j G
N2} and {Yj, j G N2} on a richer probability space such that the distribution
of {Xj,j G №} is not changed and {lj,j G №} are independent centered
normal random variables with ΕΥ? = σ2, and further
P{h
1/2
£ (Aj-Yj)|>a,}<
Oil.
Obviously we have Σι оц < oo, which implies
Ι Σ
}еНф(1)\Аф(1)
£ (X,-yj)| = O(a,fcJ[20) a.s.
(11.1.27)
(11.1.28)
Note that I2 = U(t(D(1)t(2)(2))<u<ic-fifu\Au. Take tk G ϋβ such that
*k )(1) ^ 'if)(1)'*k2)(2) ^ 42)(2)· Then for еуегУ A G Λ and some *" > °
we have
I^Ktk^-jj.n^nCjKAj-Yj)
< Σ | Σ м-*)
(i^1)(l)42)(2))<U<k J<E#u\au
< Σ
*k
V-(i)<k,t/>j(0>ti!)(i),i=i,2
а'лУ(/)
< c\tk\
1/2-t"
a.s.
(11.1.29)
For the random field {Yj} which defined in (11.1.27), we define K' like

300
Chapter 11 Strong Approximations for Mixing Random Fields
and
Xj, then for tk < η < tk+i, A e Awe have
|£|nAnCj|(X,-yj)|
j
<\^\nAnCs\(Xs-X!)
J
J j
where
I^MnCjKxj-y/)
j
< |Σ Ι Μ П Cj| - \nA(2-b) П Cj| \(Χ! - Υ/)
j
+ \£ Ι |пА(2-ь) П Cj| - \пА(2~а) П Cj| \(Χ! - У/)
J
+ \£ Ι |ηΑ(2-) η σ,| - Μ(2-) n Cj| |(Xj - У/)
j
+ \£ \(tkA(2-°)\(tkA(2-a))* П CjKX] - У/)
j
+ |£|(ikA(2-«))»nC'j|(Xj-yj)
j
(11.1.30)
|£|(ikA(2-e)),nCj|(Xj-Yj)
j
^I^Kik^-^n/xnCjKXj-yj)
j
+ |£ |(ikA(2—)), П/2П Cj|(Aj - Yj)
j
+ |^l(^(2-a))*n/3ncj|(xj-yj)
(11.1.31)
It is clear that Lemmas 11.1.2-11.1.8 also hold true for the independent
normal random field {lj,j EN2}. Then Theorem 11.1.1 is proved from
these lemmas and (11.1.29)-(11.1.31).

11.2 Strong approximations of a a -mixing random fields
301
11.2 Strong approximations for a-mixing random fields
The strong approximation results for α-mixing random fields were
given by Strittmatter (1990) for a smaller class of sets with Ν(ε) < ce~u
for some и > 0 and the condition "n G Gp" was left out.
Theorem 11.2.1. Let {Xj,j G Nd} be a weakly stationary a-mixing
random field of Rq-valued random vectors with EX$ = 0. Put ||Xj||2 =
(Χ?χ Η h X?) and suppose that there are positive constants C{ > 0, г =
1, · · · ,4, such that
ЕЩ\\2+6 < Cu for every j G Nd and some δ > 0, (11.2.1)
a(t) < C2t~s for some s > 1 + 2(17/2)^-7(1 Λ δ), (11.2.2)
£s — 4 — 2δ
Ν (ε) < C3e~u for some и < —— — - 2, (11.2.3)
α(4 + о)
6(e) := sup{n-d\((nA)n£ Π (ηΑ°)ηε)\ : A G (J Д,, η > 1/ε}
Τ7>0
< C4 ε*1 /or some 0 < h < 1. (11.2.4)
Рг/*
r(j) = Cov(Xj,Xj).
Then the series
d
jgz·
converges absolutely and Τ is a non-negative definite matrix. Furthermore,
without changing its joint distribution, we can redefine the field {Xk,k G
Nd} on a richer probability space together with a field {lk>k £ Nd} of i.i.d.
centered Gaussian random vectors with covariance matrix Τ such that for
some 7 = j(u, s, d, /ι, 5, Ν(ε)) we have
sup I ]T |nAnCj|(-Xj - Yj)| = 0(ηά/2~Ί) a.s. (11.2.5)
The proof of Theorem 11.2.1 will need the following lemmas.

302
Chapter 11 Strong Approximations for Mixing Random Fields
Lemma 11.2.1. Let {Xh j G №*} be an α-mixing random field with
£Xj=0, £||Xj||2+6 < CX and
OO
C0 = Σ га-га(г)6К2+6) < oo. (11.2.6)
r=l
Then for 0 < dj < 1, j G Nd, where only finitely many j such that dj φ 0,
we have
Ε\\Σ dixsf ^ c Έ dh
jGNd JGNd
w/iere С = (1 + ^З^С^2^.
Proof. By Lemma 1.2.4 we have
||£AjXk|| < 10a(d(j,k))i/(2+6)||Xj||2+5||Xk||2+i
= 10aKJ,k))i/(2+*)C12/(2+5),
where d(j,k) = maxi<i<d |ji — fct-|. Lemma 11.2.1 is proved.
Lemma 11.2.2. Letf {^j,j G Nd} be α weakly stationary a-mixing
random field with EX$ = 0, ||-Xj|| < Μ < oo and satisfying (11.2,6).
Let Di,Z)2,···, Dq,Dk — (n& — 2р1,щ], be mutually disjoint cubes with
nk G 2pNd for some fixed ρ G N. Le* 0 < dj < 1, j G Nd. Put
D= \jDk Fk= £ dj, F=£dj.
fc=l jGDfcHN^ j££>
Then for every К > 0 we have
^{||E^j||>2"+1^}
< c{K-2M2F2a{p)+p2dC-2MAa{p)
+ exp(-K2/(8CF)) + exp^-^tf/M^),
where С is defined as in Lemma 11.2.1.
This lemma is a generalization of Theorem 4 in Philipp (1984). The
proof follows closely by his lines, and hence is not given here.
Proof of Theorem 11.2.1.
Prom (11.2.3) it follows that there exist positive numbers r and ζ such
that ζ < \ — }^ц and
u < .С'~' - 2. (11.2.7)
«2 + 5T«)

11.2 Strong approximations of a a -mixing random fields
303
Define for j G
Ц = Х>1(\\Х>\\<\з\(1+тШ2+%
Aj - Aj - il/Aj, Aj - Aj - Aj.
For a constant β specified later define
m
ф{т) = ]Г ^ m G N. (11.2.8)
fc=i
Assume that m and η are linked by
ф{т) < η < ф(т + 1). (11.2.9)
For г = (γι, · · ·, r<i) G Ν01, write
Rr = {(νι, · · ·, u<*) € i?J. : ψ(η) <Vi< ψ(π + 1),t < d}.
For А С R% П Bd let
Λ»= (J Д,.
йгСА
and for AeBdn [0, l]d
J
5η(Λ) = χ;|ηΛησι|Χί,
j
j
We introduce the following result of Berkes and Morrow (1981) without
proof.
Proposition 11.2.1. Let {£j,j G Nd} be a weakly stationary
admixing random field with ££ = 0, £||<j||2+* < oo, a(t) = o(t-d(1+e)(1+2/*)).
Denote
d
οθ= f)tieNd:u^Uiih o<0<i.
к=1 1фк
Then the series
a2 = u# + 2£>6£j

304
Chapter 11 Strong Approximations for Mixing Random Fields
converges absolutely. Without loss of generality, assume that σ2 = 1.
Furthermore without changing the distribution of {£j}, we can redefine
the sequence {£j,j G Nd} on a richer probability space together with a
random field {?7j,j G Nd} of i.i.d. centered Gaussian random variables with
Ε η? = 1 such that for any η G Gq we have
sup
Σ & - Σ %ll = °(H1/2"A) a-s-
(11.2.10)
l<m< n"k<m k<m
where λ > 0 depends only on the field {£j}.
Now we take θ = l/(8d — 1) and denote
G = G1/(M-i), L = {r G Nd : (φ(π), ■■·, ф(га)) € G}.
By Proposition 11.2.1, we have a random field {Yj, j G Nd} of i.i.d. centered
Gaussian random variables with BY? — 1 such that
Σ ||Σ№-^)| = °(^/2"71) a-s-
reL jeRr
Уг,ф(г{)<п
for some 71 > 0. Denote
i5' = ^(lli3ll<lil(1+T)/(2+i)).
Tn(A) = J2\nAnCi\Yh
J
j
i/;(A) = ^l(nA\(n^)nCj|y/.
j
Then for к > 0(specified later on) we have
|£Mncj|(Xj-Yj)|
J
= \\Sn(A)-Tn(A)\\
< \\Sn(A) - Sn(A)\\ + \\SQ(A) - S n(A(n-K))\\
+ \\Vn(A(n-«))\\ + \\Tn(A)-T>n(A)\\
+ \\T'a(A) - T'n(A(n-«))\\ + \\K(A(n-«))\\
+ Σ (\\Х}-Х}\\ + \\У*-Г}\\)
l<j<nl
+ |^l(nA(n-'c))*ncj|(xj-yj)|
(11.2.11)
=: hi + I\2 + Лз + hi + /22 + hz + h + h- (11.2.12)

11.2 Strong approximations of a a -mixing random fields
305
Prom Lemma 11.1.2 it follows that
/n V hi V h = 0(nd/(2+V). (11.2.13)
From Lemma 11.1.3 it follows that
I\2 V /22 = 0{ndl2-^) (11.2.14)
where 72 = к - d(\ + r)/(2 + δ) - d/2 > 0.
In order to estimate /13,/23 and I4, we need the following lemmas.
Lemma 11.2.3. There exist d\\ + |+f ) < к < d, 7', 7" > 0 suc/i
</ia£ /or anj/ η G N we /ιουβ
Ρ{διιρ(||^η(Α(η-κ))|| : Λ(η~Λ) G Λ(η"")) > cnd/2-7'} < сгГ1^".
Proof. Put ρ = [n<], for fixed Л G BdП [0, l]d cover (nA)\(nA)* with
cubes in i?+ with length of edges equal to 2p and of the form explained in
Lemma 11.2.2; as there let D be the union of these cubes. For j G Nd let
dj = |((ib4.)\(ib4)*)nCj|. Then by (11.2.4) we have
F= Σ^ = |(ηΛ)\(ηΑ),|
< ndb((ip(m + 1) - ф(т))/п) < cnd-h(ip(m + 1) - ^(m))h
< cnd-hnhW+V = cnd-\ (11.2.15)
where 7 = Η/{β + 1). There is a 7' > 0 such that 7' < 7/2 and
τΜΙ-ϊϊΗ)· (11·2·16)
Then by Lemma 11.2.1, (11.2.15) and Lemma 11.2.2, we have
Р{\\ЩА)\\ > cndl^'}
< c(n-d+2'('M2F2a(p)+p2dM4a(p)
( nd~2A ( οηάΙ2-Ί\χ

306
Chapter 11 Strong Approximations for Mixing Random Fields
Since Μ = 2И1+г)/(2+*), ρ ~ η<, we have
P{\\Vn{A)\\>cndl2^'}
, 1 + r
f -ά+2Ί'+2ά- -+2d-27-<s
<c[n 2 + ό
1+r
2<*<+4d——-<s
+ n 2 + ό
, 1+r
/ / \ / d/2-~f'-d c-d£\\
+ expi -cn~21 +7J + expi -en 2 + ό J J
, 1 + r.
<сп К 2 + 5У. (11.2.17)
There exist θ and к with 0 < θ < ζβ - 1 - d(l + ^ +J )> d(s + 5+£) <
к < d and 0/я = u. We multiply both sides of (11.2.17) by N(n~K) <
cnKU = en6 and obtain
P{ sup \\ЩА(п-"))\\ > cndl2-^')
L A(n~«) }
1 + r
θ-ζ8+2{1+ά—--)
<cn * + о
-l-V
= en 7 .
The proof of Lemma 11.2.3 is completed.
Lemma 11.2.4. If in (11.2.8) β > 6d then
P{ Σ Ι Σ X i\\ > nd/2"1/16} < cm~2, (11.2.18)
г£Ь,<ф(п)<п jeR r
where η and m are linked by (11.2.9), hence by the Borel-Cantelli lemma
Σ | Σ Xi\ = 0(ndl2-l'w) a.s. (11.2.19)
т£Ь,ф(п)<п jeЯ r
Proof. For r $. L with ф{г{) < η, г = 1, · · ·, d we have by definition
of L for some г < d, ф{п)ы < Π?=ι Φ(η) < nd, hence <ψ(η) < η1/8. And
therefore
Card(R r П Nd) < n1/8 · nd~l = nd-7/8.
Furthermore we have
Card{< : t G N,^(t) < n} < Card{i : i G N,ci^+1 < n} < cn1/(/3+1),
Card{r:r €Nd,V(ri) < n,i = 1, · · · ,d} <
en
<*/(/?+!)

11.2 Strong approximations of a a -mixing random fields
307
By Lemma 11.2.1 and the Chebyshev inequality this implies
p{ Σ || Σ *j|>nd/2-1/16}
r£L,<0(74)<n jeR г
<cnd№+1) max P\\\ V X Jl > cn*~™~eTi
< СП^1 · η 8 α+8 + /3+1
3 ■ 3d
= СП 4^/3+1
< cm~2.
Lemma 11.2.4 is proved.
Prom Lemma 11.2.3 it follows that I13 V /2з = 0(ηά/2~Ί'). Finally
since
'4< Σ Ι Σ &}-Υ})\\
veL^(ri)<n jeR г
+ Σ \\Σ(χί-γί)
г£Ь,ф(г{)<п jeR г
we have I4 = 0{nd'2-^) a.s. by Lemma 11.2.4 and (11.2.11). Therefore
the proof of Theorem 11.2.1 is completed.

Part IV Statistics of a Dependent Sample
The limit behavior of various kinds of statistics with a dependent
random sample have been studied by many authors since the sixties. We shall
introduce some of them in this part. We first introduce weak convergence
and strong approximations of an empirical process with a mixing
dependent sample in Chapter 12. The limit behavior of U-statistics, estimations
of error variance in a linear model and estimations of density function with
a mixing dependent sample are discussed in Chapter 13. We investigate
in Chapter 14 the asymptotic fluctuation behavior of sums of other kinds
of dependent random variables, such as lacunary trigonometric series, a
Gaussian sequence and the additive functional of a Markov process.

Chapter 12 Empirical Processes
Let {Xn,n > 1} be a sequence of random variables with a common
distribution function F. Then the empirical distribution function Fn of
Xu---,Xn is defined by Fn(t) = η"1 Σ?=ι J(xi < *)> ~°° < x < °°· The
nth empirical process βη is defined by
Pn(t) = y/n(Fn(t) - F(t)), -oo < t < oo. (12.0.1)
If F is continuous, then Un := F(Xn) for all η > 1 are uniformly
distributed over [0,1]. The corresponding empirical process is
<*n(t) = y/n(En(t) - t), 0 < t < 1, (12.0.2)
where the empirical distribution function
1 n
£?n(i) = Fn(mvF(t)) = ~Tl(Ui<t), 0<t< 1. (12.0.3)
ηΤΞ[
Thus, in term of JJ{ = F{X{), г = 1, · · ·, η, we have for any continuous F
{/3n(invF(t)), 0 < t < 1} = {an(t), 0 < t < l}, η = 1,2, · · ·. (12.0.4)
This implies that all theorems proved for an will hold automatically for βη
as well, simply by letting у = F(x) in (12.0.4). So we shall mainly concern
with uniform empirical process an in this Chapter.
In the independent case, it is well-known that
an => В as η —► oo (12.0.5)
where В is a Brownian bridge. The strong approximations of {c*n(·)} by
a sequence of Brownian bridges were discussed by Komlos-Major-Tusnady
(1975). They showed that without changing the distribution of {αη(£), η >
1}, one can redefine the {an(£)} on a richer probability space together with
a sequence {Bn} of independent Brownian bridges such that
sup \an(t) -Bn(t)| = 0(n~1/2 log n) a.s.. (12.0.6)
o<t<i

312
Chapter 12 Empirical Processes
For a strictly stationary sequence {Un,n > l}, the conditions, under
which (12 0.5) and (12.0.6) also hold true, have been given by many
mathematicians. This chapter is organized as follows. We shall introduce the
best results of weak convergence of empirical processes when the sample
is mixing dependent in Section 12.1. The weighted weak convergence for
empirical processes of a-mixing and p-mixing sequences will be established
in Section 12.2. The strong approximations for empirical processes with
a mixing sample by Gaussian processes will be introduced in Section 12.3
and the moduli of continuity of empirical processes when the sample is
mixing dependent will be studied in Section 12.4.
12.1 Weak convergence
Let {Un,n > 1} be a sequence of strictly stationary random variables
uniformly distributed over [0,1], denote
oo
R(s,t) = sAt-st + Y^ E{[I(Uk <s)- s][/([/x < t) - t]
+ [Wi < a) - s][I(Uk < t) - t}}. (12.1.1)
When {Un} is α-mixing, the series in (12.1.1) converges absolutely if
Σηα(η) < oo by Lemma 1.2.1.
The weak convergence of empirical processes {an(t),0 < t < 1} of
{Un} has been discussed by some scientists. For an insightful view of this
subject we refer to Billingsley (1968), Sen (1971, 1974), Yoshihara (1975,
1978) and Shao (1986), etc, until now the best results are due to Shao
(1986).
12Λ Λ Weak convergence for a φ-mixing sample
Theorem 12.1.1. Let {Un,n > 1} be α sequence of strictly stationary
φ-mixing random variables uniformly distributed over [0,1], and let
an(t) = V^(En(t) - t). (12.1.2)
If the series of (12.1.1) converges absolutely and
oo
$>х/2(2") < oo, (12.1.3)
n=l
then we have
an=>Y inD[0,1],

12.1 Weak convergence
313
where Υ = {Y(t),0 < t < 1} is a Gaussian process with EY(t) = 0,
EY(s)Y(t) = R(s,t).
Proof. For any given t £ [0,1],
αη(ί) = -7=Σ(/(Χ*^')-*)·
У71 4.-1
By Corollary 4.1.1, an(t) converge to Y(t) in distribution and it is easy to
see that the finite dimensional distributions of an converge to the
corresponding finite dimensional distributions of Y.
In order to prove the tightness of {an}, we need only to show that
for any ε > 0, η > 0 there exists a <5,0 < δ < 1, such that for any
5,0 < 5 < 1 — 5, and large n
P{ sup \an(t) - an(s)\ > 4ε} < ηδ. (12.1.4)
s<t<s+6
For 0 < 5 < t < 1 denote & = (1(Щ < t) - t) - {1{Щ < s) - s) = I(s <
Ui<t)~ (t-s). We have
Εξι = 0, |6| < 1, Eg = t - s - (t - s)2 < t - 5.
From Lemma 2.2.2 and Lemma 2.2.8 it follows that
E\an(t)-an(s)\4<±E(JT,tj)4
3 = 1
<C(t-s)2 (12.1.5)
for some С > 0. Let ρ be a positive number such that ε/π < p. Consider
the random variables an(s + ip) — an(s + (г — l)p), г = 1,2, · · ·, m. From
Theorem 12.2 of Billingsley (1968) we have
P{ max \an(s + ip) - an(s)\ > \\ < ~rjm2p2 (12.1.6)
^0<i<m J ελ
where constant К depends only on φ(·).
For 5 < t < s + p, we have
\<*n(t) - an(s)\ < \an(s +p) - an(s)\ + pyfti, (12.1.7)
which implies
sup \an(t) - an(s)\
s<t<s+pm
< sup {\an(t) - an(s + ip)\ + |an(5 + ip) - an{s)\]
s<t<s+pm
< 3max \an(s + ip) — an(s)\ +py/n. (12.1.8)
i<m

314
Chapter 12 Empirical Processes
If ε/η<ρ< ε/ν/η, from (12.1.6) and (12.1.8) it follows that
{K
sup \an(t) - an(s)\ > 4ε} < -,т2р2. (12.1.9)
s<t<s+pm J ε
Take δ such that Κδ/ε5 < η. For large n there exists an m such that
(δ/ε)^η <m< (δ/ε)π and mp = δ. It follows from (12.1.8) and (12.1.9)
that (12.1.4) holds true. The proof of Theorem 12.1.1 is completed.
Corollary 12.1.1. Let {Un,n > 1} be a sequence of strictly stationary
p-mixing random variables uniformly distributed over [0,1]. If the series
in (12.1.1) converges absolutely and Υ^=ιΡ{^η) < oo, then an =>Y.
Remark 12.1.1. In the case of a p-dimensional sample (p > 2), the
result of Theorem 12.1.1 is also true.
Remark 12.1.2. The Glivenko-Cantelli theorem, i.e. almost sure
convergence of empirical processes of mixing sequences, is an immediate
consequence of the results in Sections 8.3—8.5. For example, from
Corollary 8.3.1 we have
Proposition 12.1.1. Let {Xn, > 1} be a sequence of (^-mixing
random variables with a common continuous distribution F(x). Then for any
θ > 0, ε > 0
oo
Σ P{\Fn(x) - F(x)\ > еп-^2+в} < oo,
71=1
that is to say the rate of convergence in the Glivenko-Cantelli theorem is
given as follows:
\Fn(x) - F(x)\ = ο(η-χ/2+θ) a.s.
When the sample is /o-mixing or α-mixng, there is the similar results
under certain conditions.
12.1.2 Weak convergence for an α-mixing sample
Theorem 12.1.2. Let {Un,n > 1} be α sequence of strictly stationary
α-mixing random variables uniformly distributed over [0,1] and
a{n) = 0(n-r) r > 2. (12.1.10)
Then an => Y.
By the following lemmas, from the proof of Theoem 12.1.1 it follows
that Theorem 12.1.2 holds true.

12.1 Weak convergence
315
Lemma 12.1.1. Let {ξη·>η > 1} be a sequence of strictly stationary a-
mixing random variables with |£i| < 1 a.s., Εξι = 0, Εξ\ = τ, Ε\ξχ\ = 2τ
and satisfying (12.1.10). Put Sn = Σ"=1 fj. Suppose that r > 2 is a non-
integer and m = 2k, where к is an integer, satisfies r — 1 < ra < r + 1.
Then we have
(m-2)/2
E\Sn\m<c Σ (ηί+1τ"1+*—^^ (12.1.11)
t=0
where [r] - 1 < rB\ < r - 1, вк = 0\ ~ (k - l)/r, к = 1, · · ·, [r].
Proof. By (12.1.10) and the definition of 0fc, we have
oo
J2(i + l)*-1*1"^*) < oo, (12.1.12)
t=0
for к = 1, · · ·, [r]. Let Ση(*Λ be a summation for ii, · · ·, i& > 0, ii + · · · +
ik < n, and Ση/м be a summation for ii, · · ·, %k > 0, г\Л h г& < η and
i\ = maxi<j<fc ij, where / = 1, · · ·, fc; fc = 1,· · ·, [r]. Denote
Ση(*)= Ση(Λ)ΐ£7^ι^+« •••^ι+···+ΰΐ·
Note that for any even d, by the stationarity, we have
E\Sn\d < άΙηΣ^^Εζοξ* ■ •·ξίι+...+ίΛ-1\. (12-1.13)
From Lemma 1.2.5 and (12.1.12) we have
Ση(1)|^ο£ύΙ < 6 Σ ах-^{чУ- <ств\ (12.1.14)
Ип(2)\Е^г^г1+г2\ < {Ση(2) + Έη(2)}\Ε^ξήξή+ί2\
< 6 £ (ή + lja1"*» (ixjr* + 6 £ (г2 + 1)^-^(12)7^
ii=0 гг=0
<Οτ"2. (12.1.15)
Next we prove
Ρ-1)/2]
En(k)<c Σ nV*+*-« (12.1.16)
г=0

316
Chapter 12 Empirical Processes
for к = 1,2,···,[γ] by the induction. Prom (12.1.14) and (12.1.15) it
follows that (12.1.16) holds true for к = 1,2. Assume that (12.1.16) holds
true for к — 1, 2 < к — 1 < [r], we show that (12.1.16) also holds true for
k. Note that
у <yw +... + y{k)
E(1i<6E(?,y_6,fc(4)^
η
ii=0
Similarly, ^jj^ < cr**. For 2 < / < A; - 1,
F(/L<6r('L«1"eb(ii)^b
=: h + h-
We also have I\ < ствк. By the induction and stationarity we have
[{1-2)12} t(fc-'-D/2]
г=0 j=0
< c y^ n*+i+ir(«+j+i)^i+^-2(i+i+i)
0<i<[bl],0<j<[*^=l]
2 J 5 ^^J^L 2
[(*-l)/2]
г=1
< С Σ nV*i+'*-2i
Thus (12.1.16) holds true for A; = l,2,---,[r]. From (12.1.13), (12.1.16)
and noting m — 1 < [r] we complete the proof of Lemma 12.1.1.
Lemma 12.1.2. If the conditions of Lemma 12.1.1 are satisfied with
2 < r < 3, then
ES* < c(nA~r + η2τ2θι). (12.1.17)
Proof. From Lemma 1.2.5 and (12.1.14) we have
21=0

12.2 Weighted weak convergence
317
Similarly J2%) < <™3_Γ· And by (12.1.16) we have
<c(n3-r + nr2^).
Combining these inequalities with (12.1.13) yields (12.1.17).
Now Theorem 12.1.2 can be proved along the similar lines to that of
Theorem 12.1.1 by applying Lemma 12.1.2 instead of Lemma 2.2.2 and
Lemma 2.2.8.
Remark 12.1.3. In the case of a p-dimensional sample (p > 2), by
using Lemma 12.1.1, the result of Theorem 12.1.2 is also true if
r > ρ + 1 when ρ is even;
r > p2/(p — 1) when ρ is odd.
Remark 12.1.4. The weak convergence of partial-sum processes and
empirical processes with random indexes has been discussed by some
scientist. For this subject we refer to Renyi (1958, 1960), Billingsley (1968 §17),
Aldous (1978) and Lu (1984), etc. We enumerate only a result without
proof for the weak convergence of empirical processes with random indexes
as follows:
Let {an,n > 1} be a sequence of empirical processes as in Theorem
12.1.2 and let {rn,n > 1} be a sequence of positive integer-valued random
variables on the same probability space. Suppose that an => У, where Υ
is a Gaussian process and {rn} satisfies
# Ρ
Τη/Π ► T,
where r is a positive random variable. Then we have
a
Tn
12.2 Weighted weak convergence
Let q be a positive weight function on (0,1), i.e. inf^^!-^ q(t) > 0
for all 0 < δ < 1/2, and define the weighted uniform empirical processes
as {ctn(t)/q(t),0 < t < 1}. When {Un,n > 1} is a sequence of
independent random variables uniformly distributed on [0,1], the weighted weak
convergence of empirical processes has been intensively studied in recent
years. For an insightful view of this subject we refer to M. Csorgo, S.
Csorgo, Horvath and Mason (1986a), Shorack and Wellner (1986) and

318
Chapter 12 Empirical Processes
Csorgo and Horvath (1993), etc. We restate a theorem of M. Csorgo, S.
Csorgo, Horvath and Mason (1986a) as follows. A shorter and more direct
proof was given by Csorgo and Horvath (1993 Chapter 4).
Theorem 12.2.1. Assume that q is positive and continuous on (0,1),
and is nondecreasing in a neighborhood of 0 and nonincreasing in a
neighborhood of 1. Then
I(q, A) := £ j^^—j exp(-\q2(t)/(t(l - t)))dt < oo (12.2.1)
for all λ > 0 if and only if, as η —> oo
an/q => B/q in D[0,1]. (12.2.2)
There are only a few studies concerned with the weighted weak
convergence for empirical processes of a dependent sequence. In the latter
case the limit process is changed from being a Brownian bridge, due to the
appearence of covariances among observations (cf. (12.1.1)).
Shao and Yu (1995) studied the weighted weak convergence for
empirical processes of strictly stationary observations under mixing and
associated dependence assumptions. We only introduce the case of mixing
dependence here. First we give the following basic theorem.
Theorem 12.2.2. Let {Un,n > 1} be a strictly stationary sequence of
uniform-[0, 1] random variables. Assume that for all 0 < s,t < 1 and
η > 1 we have
(Al) E\an(t) - an(s)\P < Cx{\t - s\^ + n~^l2\t - s|ri) for some
Cx > 0,p > 2, pi > 1, 0 < ri < 1 andp2 > 1 - rx;
(A2) E(an(t)-an(s))2 < C2\t-s\r2 for some C2 > 0 andO < r2 < 1.
// we have
an=>Y in L>[0,1] (12.2.3)
with the Gaussian process Y(-) defined as in § 12.1, then
an/q=>Y/q mL>[0,l], (12.2.4)
where q is a weight function such that for some С > 0 and β > 1/2
q{t) > C(t(l - *)Hlog l/(t(l - ΐ)))β for all0<t<l (12.2.5)
and
. (Pi П +Р2 Г2\ nooa\
μ = ταιη[—,—■ ,—). (12.2.6)
V ρ ρ + p2 2 /

12.2 Weighted weak convergence
319
Remark 12.2.1. By using a standard argument (cf. Theorem 12.2
and (22.18) in Billingsley 1968), one can easily verify that {an(£),0 <t<
1} is tight by condition (Al). Hence to show (12.2.3), one needs only to
prove that any finite dimensional distribution of {an(t)} converges to that
of {^(t)} and the series in (12.1.1) converges absolutely.
Remark 12.2.2. The weight function q used in Theorem 12.2.1 is
usually called a Chibisov-O'Reilly weight function. If we write q(t) =
(t(l - t)loglog(l/(t(l - £))))1/2#(£), then, necessarily, g(t) -► oo as t -► 0
or t —► 1. Thus the weight function q in (12.2.5) can be compared to
a Chibisov-O'Reilly weight function by taking μ in (12.2.6) close to 1/2
or exactly 1/2 for properly chosen p,p\,Pi,r\, an(i r2· In fact, Theorems
12.2.3 and 12.2.4 below show this possibility of taking μ = 1/(2 + ε) for
some ε > 0 in the case of mixing sequences. In particular our sharpest
rate with μ = 1/2 is obtained for p-mixing under a stronger mixing decay
rate. In most cases, however, μ < 1/2. We note in passing that if for a
general weight function q we have /0 l/q2(t)dt < oo, then we have (12.2.1)
as well for all λ > 0, i.e., q is then necessarily a Chibisov-O'Reilly weight
function.
Remark 12.2.3. If μ = (rx + рг)/(р + P2) < min(pi/p,r2/2) in
(12.2.6), then from the proof of Theorem 12.2.2, one can relax the
restriction on β from β > 1/2 to β > l/(p + P2) = (1 — μ)/(ρ — r\). Moreover,
in the case of μ > l/(p+l — Γχ), one can use a simple sufficient condition
Jo l/q1/fI(t)dt < 00 to replace (12.2.5).
A direct application of Theorem 12.2.2 is to obtain weak convergence
for integral functionals of an. For example, we consider the integral
functional
Δη(ί) = / an(s)dQ(s) = [ fin(invF(s))dQ(s), 0 < t < 1
Jo Jo
and its approximating Gaussian counterpart
A(t) = [ Y(s)dQ(s), 0 < t < 1, (12.2.7)
Jo
where Q(s) = invF(s) is the quantile function of distribution function F
of X (recalling (12.0.4)). The function A(t) plays a central role in weak
approximation theory for empirical total time on test, mean residual life,
empirical Lorenz and Goldie concentration processes which are of interest
in reliability and economic concentration theories, (cf. e.g., M. Csorgo, S.
Csorgo, Horvath and Mason 1986b).

320
Chapter 12 Empirical Processes
Corollary 12.2.1. Under the conditions of Theorem 12.2.2, if
[\t(l - t)r (log l/(t(l - t))fdQ{t) < oo, (12.2.8)
then
Δη=>Δ in £>[0,1].
Remark 12.2.4. Let F be the distribution function of a random
variable X. Then condition (12.2.8) is sightly stronger than the existence
of the (l//i)-th moment of X. This is not necessarily true conversely, but
E\X\ll»(\og(l + |χ|))(ι+/3)/μ+* < oo5 With any δ > 0, implies (12.2.8).
Theorem 12.2.2 enables us to establish weighted weak convergence for
empirical processes of a stationary mixing sequence.
Theorem 12.2.3. Let {Un,n > 1} be a strictly stationary a-mixing
sequence of uniform-[0, 1] random variables. If
a(n) = 0(η-θ-£) (12.2.9)
for some θ > 1 + л/2 and ε > 0, then we have
an/q=>Y/q in D[0,1]
for q satisfying q(t) > C(t(l - ί))(1~1/^)/2 for some С > 0.
Theorem 12.2.4. Let {Un,n > 1} be a strictly stationary p-mixing
sequence of uniform-[0, 1] random variables. Suppose that the series in
(12.1.1) converges absolutely. If
oo
]T,9(2n) <oo, (12.2.10)
n=l
then for any ε > 0 we have
an/q=>Y/q in L>[0,1]
for q satisfying q(t) > C(t(l - t))1/^) for some С > 0.
Iff in addition,
oo
Σ PVp(2n) < oo (12.2.11)
n=l
for some ρ > 2, then we have
an/q => Y/q in D[0, l]

12.2 Weighted weak convergence
321
for q satisfying q(t) > C(t{\ - i))1/2(bg l/(i(l - ί)))β for some С > 0 and
β > 1/2.
Corollary 12.2.2. Under the conditions of Theorem 12.2.3, if
Γ Ixf^-WdFix) < oo,
J — OO
then
Δη=>Δ inD[0,l].
Corollary 12.2.3. Let {Un,n > 1} be a strictly stationary p-mixing
sequences of uniform-[0, 1] random variables. If (12.2.10) holds and for
any ε > 0 we have
/oo
\x\2+edF{x) < oo,
-OO
then
Δη=*Δ.
//, in addition, (12.2.11) holds and
i\t(l-t)Y/2(logl/(t(l-t))fdQ(t)
JO
then we have
Δη =>Δ in £>[0,1].
In order to prove Theorems, we need the following lemmas.
Lemma 12.2.1. Let {£;,г > 1} be a sequence of random variables and
let T{ — &(£j)j < i). Then, for any ρ > 2, there exists a constant D = D(p)
such that
^|Σ^Γ<ΰ((Σ^)Ρ/2+Σ^ιρ+ηί'-1Σ^(6ΐ^-1)Γ
г=1 г=1 г=1 г=1
+ nP/2-1 jr В|£7(е?|^_!) - Ε$\ρΙ2) ■ (12.2.12)
i=l
Proof. Let r)i = ξ{ — E^Fi-i) for 1 < г < п. Then, {^,^-ι, Ι <
г < η} is a martingale difference sequence. By the well-known Burkholder

322
Chapter 12 Empirical Processes
(1973) inequality, there is a D = D{p) < oo such that
E\±m\P < D{E(±E{ri\^))P'2+ ±Ε\ηί\ή
i=l i=l i=l
<2^((±Εξ!)Ρ/\±Ε\ξ^
г=1 г=1
,р/2ч
+ ^(ΣΙ^2Ι^-ι)-^2Ι)ρ/)
i=l
< _
г=1 г=1
+ ηΡ/2-1Σ£|^|^_1) -Ε&\ρΙ2). (12.2.13)
г=1
On the other hand, it is easy to see that
E\^i\P <2p(E\J2m\P + np-1J2E\E^i\^i-i)\P)-
i=l i=l i=l
This proves (12.2.12) by the inequalities above.
We now develop a Rosenthal-type inequality for an α-mixing sequence
which is of its own interest.
Lemma 12.2.2. Let 2 < ρ < r < oo, 2 < ν < r and {Xn,n > 1} be
an α-mixing sequence of random variables with EXn = 0 and \\Xn\\r < °o·
Assume that
a(n) < Cn~e (12.2.14)
for some С > 0 and θ > 0. Then, for any ε > 0, there exists а К —
K(e,r,p, г>,#, С) < oo such that
E\Sn\P < K({nCnyl2max \\ХД\* + п(р-(г-р)9/г)у(1+£) щах цх.цр\
V г<п г<п /
(12.2.15)
where Cn = [Ул=0{г + ΐγ^ν~Ζ)α(ι)J . /η particular, for any ε > 0,
^|5п|р<^^/2тах||^||^ + г11+£тах||Х,||^, (12.2.16)
V г<п г<тг /
if θ > ν/(ν - 2) and θ > (ρ - l)r/(r - ρ), and
E\Sn\p < Κηρ/2 max ||Χ;||£, (12.2.17)
г<тг

12.2 Weighted weak convergence
323
*/0>pr/(2(r-p)).
Proof. For the sake of convenience of statement, we assume that
{X, Xn,n > 1} is a strictly stationary α-mixing sequence. By a result of
Rio (1993), there is Dx = Dx(v) such that
ESl < D^CnWXWl (12.2.18)
We shall prove (12.2.15) by induction on n. Suppose that for each 1 < к <
η,
E\sk\p < ^^((ЛтСл,)^/2!!^^!!^ н- л:^-^-^>^/т*)х/(1+е>||^гц^). (12.2.19)
We now prove that (12.2.19) is still true for к = n. Let 0 < a < 1/2 that
will be specified later and let m = [an] + 1. Define
пЛ(2г-1)т пЛ2гт
ξι = Σ Χ3 aild ГЧ= Σ XJ
j=2(i-l)m+l j=(2z-l)m+l
for 1 < г < кп := [η/(2m)] + 1. Clearly, we have
E\Sn\> < 2ρ-1(^|Σ^|Ρ + ^|Σ^Π =: 2P~\h + h).
г=1 г=1
Let Ti = σ(^·, j < г). It follows from Lemma 12.2.1 that there is Д2 such
that D2 > (2L>i)*>/2 and
г=1 г=1 г=1
i=l
к
=:£>2(£вд" + /11 + /12 + /13). (12.2.20)
г=1
In terms of (12.2.18), we have
hi < {DxknmCm\\X\\lYl2
< (2DinCnr/2\\Χ\\ζ < D2{nCnf'2\\Х\\Р. (12.2.21)
To estimate /13, we write
Yi = E{ei\Fi-l)-Edl

324
Chapter 12 Empirical Processes
Then, by Lemma 1.2.4
Е\ЩР'2 = ElW^sgaWYi = EdYil^-hgniYi)^ - Εξ?))
Σ ElY^-hgniYiXXjXt - EXjXt)
2(i-l)m<j,l<nA(2i-l)m
χ-л ι P~2 2
< 12 2^ a p r (m)
2(i-l)m<j,l<nA(2i-l)m
■(EW^b-W'WXjXtWrp
<12т2ар~^(т)(£;|^|р/2)(р-2)/р||Х||2,
and hence
ЕЩГ'2 < 12р/2траг-р/г(т)||Х||Р, (12.2.22)
which, together with (12.2.14), yields
/13 < kpJ42pmpal-plT{m)\\X\\p
<C2Apnpm{-p~r^lT\\X\\p
< C2Apa{-p-r)elTnp+{p~r)elr\\X\\p
< С24ра(р-г)е/Гп(р+(р~г)е/г)у(1+е)||Х||Р. (12.2.23)
Similarly to (12.2.22), we have
Ε\Ε{ξ№-Χ)\* < l2pmpax~plr{m)\\X\\p. (12.2.24)
Therefore
/12 < kpn\2pmpax-plr{m)\\X\\p
< С24ра(р-г)е/гп(р+(р-г)е/г)у(1+е)||Х||Р. (12.2.25)
Putting the inequalities above together yields
h < D2(j^E\^\p + D2(nCn)p/2\\X\\pv
i=l
+ 2C24pa(p~r^/rri^+(p~r^/r)v(1+e)||X||p).
Similarly,
h < D2(£Eh\P + Ό2{ηΟηγΙ2\\Χ\\1
i=l
+ 2C24pa^~r^/rn^+(p~r^/r)v(1+£)||X||p).

12.2 Weighted weak convergence
325
Consequently, we have
Kn
E\Sn\p < 2p~lD2{j^{EUP + ЕЫП + 2D2{nCn)pl2\\X\\p
+ AC2Apa^p~r^lTn^+^p-r^/r)^1+£\\X\\P). (12.2.26)
Now we let
a = (2p+4D2)-1/£~p/{p-2\
К = 2P+1D2(D2 + 2С24Ра^~т)е'т).
By (12.2.26) and induction hypothesis (12.2.19), we get
E\Sn\p < 2pD2(knK({mCm)pl'2\\X\\p + m(p+(P-r)e/rMi+e^x^
+ D2{nCn)pl2\\X\\p + 2C2Apa^~r)elTn^p+^~T^lT^l+^\\X\\P)
< 2pD2{nlm)K((mCn)pl2\\X\\p + ro(^(p-r)Wv(i+0||jf||p)
+ 2pD2(D2 + 2C2ApaSp-r^lr)
■ ([пСп)р'2\\Х\\р + n(p+(p-rWrW1+e\\X\\p)
< 2pD2K(a{-p-2^2{nCn)pl2\\X\\p + αεη(ρ+(ρ-τΜτ^1+εϊ\\Χ\\ή
+ (К/2) [(nCn )p/2\\X\\p + n(p+(p-rWrM1+£)\\X\\p)
< {К/2)[{пСп)р12\\Х\\р + η(ρ+(ρ-ΓΜΓ^1+εϊ\\Χ\\ή
+ (К/2) [(nCn )p/2\\X\\p + η(ρ+(ρ-ΓΜΓ^1+εϊ\\Χ\\ρ)
= к([псп)р12\\х\\р + n<p+<p-rWr)v(1+e>||x||?).
This proves that (12.2.19) remains valid for A; = n, as desired.
Proof of Theorem 12.2.2. By Theorem 4.2 in Billingsley (1968),
it is sufficient to prove that for any ε > 0
lim limsup p{ sup \an(t)/q(t)\ > ε) = 0, (12.2.27)
0->O n-»oo *·Ο<ί<0 '
limlimsupPJ sup \an(t)/q(t)\ > ε] = 0, (12.2.28)
lim P{ sup \Y(t)/q(t)\ > ε) = 0, (12.2.29)
0-»o L o<t<e >
lim P{ sup \Y(t)/q(t)\ > ε) = 0. (12.2.30)

326 Chapter 12 Empirical Processes
Note that
p{ sup |an(i)/g(i)l > ε}
^o<t<e J
oo
<ΣΡ{ sup \an(t)/q(t)\>e}
j=l θ2-ί<ί<θ2~ΐ+1 J
oo
<ΣΡ{ sup \αη(ί)\>ες(θ2-*)}.
j=l θ2-3<ί<θ2-ΐ+1 J
Hence, (12.2.27) can be rewritten as
limlimsup Pi sup \an(t)/q(t)\ >£f
^o<t<0
oo
< limsuplimsupy^ 5j?n, (12.2.31)
0_>O n->oo . = 1
where B^n = P{sup0<t<^2-i+i |αη(<)| > ες(θ2~3)}.
Put
ej = ες(θ2->), Gn = {j : η1Ι2θ2'^1 < ej/2},
Hn = {j : η1Ι2θ2'^1 > sj/2}.
It is easy to check that for any 0<s<t<s + h<l
MO - <*n(s)\ < \an(s + h) - an(s)\ + n^h. ' (12.2.32)
Hence we have for j £ Gn
Bj,n < Ρ{\αη(θ2-ί+1)\ + η1/202^'+1 > £j}
< Ρ{\αη(θ2-^)\ > ej/2}
< ceJ2(e2-JY2
< ce~2 (log(2J'/0)) ~W (12.2.33)
by (A2), (12.2.5) and (12.2.6). Hence (12.2.33) implies that
limsuplimsup Y\ Bjn = 0. (12.2.34)
e-o n^oo jeGn
Note that in the case of μ < r2/2, (12.2.34) holds also for /3 = 0.
Write
д.-д. _1 4 _eq(02->)
A-Aj'"~4nV2-4 nV2 · {12.2.35)

12.2 Weighted weak convergence
327
When j € Hn, using (12.2.32) again, we obtain
Bj,n < P{ max |αη(*Δ)| > ε,/2)
+ P\ max sup \an(t) — an(iA)\ > eJ2>
lo<i<i2-i+VA ίΔ<ί<(ί+1)Δ J
< P( max Ια„(ίΔ)| > ε,/2)
+ Ρ{ max |an((t + 1)Δ) - α„(»Δ)| + Δη1/2 > ε,/2)
< Pi max |αη(ϊΔ)| > ε,·/2}
«·1<«<β2-ί+ι/Δ "" J
+ ρ{ max |α„((ί + 1)Δ)-αη(ίΔ)| >ε,·/4|
< 3Ρ( max Ιαη(ίΔ)| > ε,/β). (12.2.36)
[-1<г<в2-з+2/А J >
From (ΑΙ) it follows that for all 0 < г < к < <92_J+2/A
E\an(kA) - an(iA)\p < Cx [{{k - i)A)p* + rT^I2{{k - ζ)Δ)Γι)
< Cx {((к - i)A)P1 + п~р*12{к - г)АГ1У
Thus, by Moricz's theorem (Lemma 4.1.2), there is a constant C,
depending only on C\ and pi, such that
Ε max |ап(гД)|р
0<ί<θ2-ί+2/Α
< θ((θ2-ί+2/Α)ΡιΑΡι+η-ρ2/2(θ2-ί+2/Α)Ατ> 1о§р(02^'+2/А))
< C4p((e2~j)pi + η-ρ2Ι2θ2-ίΑτι~ι logp(02-J'+2/A)). (12.2.37)
Since p2 > 1 - ri in (Al), l(x) = (logx)p/x-1+ri+P2 < с for all χ >
1. Thus from (12.2.5), (12.2.6), (12.2.35), (12.2.37) and the fact that
e2-i+4n1/2/ej > 8, we conclude that for j € Hn
P\ max Ια„(ίΔ)| > ε,/δ]·
•-0<i<6l2-i+2/A ~~ J >
< с£7р((02^')Р1 +n-p2/2^2-JAri-1logp(^2-J+2/A))
< се7Р((д2^")р1 + εγ-1(θ2-ήη^-τι~ρ^2 \ogp{e2-j+in1l2lej))
< cejp((e2~j)pi + ejp2(e2^)ri+p2l(e2-j+4n^2/e,-))
< ο{ε~ρ(θ2~ήρ1 +£Τρ-ρ2(θ2-ήΓι+Ρ2)
<ce~p~P2(log(2j/e)yPl3.

328
Chapter 12 Empirical Processes
This, together with (12.2.36), proves that
limsuplimsup V Bjn = 0. (12.2.38)
*-o n-oo jeHn
Note that in the case of μ < pxjp, (12.2.38) is true for β > l/(p+p2). The
proof of (12.2.27) is now completed by (12.2.31), (12.2.34) and (12.2.38).
Similarly one can prove (12.2.28).
By (12.2.3) we have for any 0 < s,t < 1
an(t)-an(s)=>Y(t)-Y(s).
Then by (A2) and Theorem 5.3 in Billingsley (1968)
E(Y(t) - Y(s))2 < liminf E(an(t) - an(s))2 < C2\t - з\Г2. (12.2.39)
η—►oo
Thus, based on the fact that {Y(t),0 < t < 1} is a Gaussian process, we
have
E(Y(t) - Y(s))4 < cE2(Y(t) - Y(s))2 < c\t - s\2r2
for all 0 < s,t < 1. Applying Theorem 12.2 in Billingsley (1968), we
can immediately get (12.2.29) and (12.2.30). This completes the proof of
Theorem 12.2.2.
Proof of Corollary 12.2.1. First we verify that {Δη(2), η > 1} and
Δ(2) are well defined on [0, 1]. By Schwarz's inequality, (A2), (12.2.6) and
(12.2.8), for 0 < t < 1,
E/±
2n{t) = I [ Ean(u)an(v)dQ(u)dQ(v)
Jo Jo
< (£ EV2(an(t))2dQ(t)f
<2r*C2(l (t(l-t))r^2dQ(tYj
< oo,
where the last inequality follows from an(0) = an(l) = 0, E(an(t))2 <
C2tT2 for 0 < ί < 1/2, and E(an(t))2 < C2(l - t)r2 for 1/2 < t < 1.
Similarly, using (12.2.39) in conjunction with (A2), we have
EA2(t) = [ f EY(u)Y(v)dQ(u)dQ(v)
Jo Jo
<{j^E"\Y{t)YdQ{t)f
< 2r2C2(f (<(1 - t))T*'2dQ(t)f < oo.

12.2 Weighted weak convergence
329
This shows that {Δη(<), 0 < t < 1; η > 1} and {Δ(<), 0 < t < 1} are square
integrable processes. Now we have for any θ > 0
sup |Δη(ί)| < sup |αη(ί)/9*(ί)Ι / q*(t)dQ(t)
o<t<e o<t<e Jo
and
sup |Δη(1)-Δη(ί)|< sup \an(t)/q*(t)\ f q*(t)dQ(t),
1-θ<ί<1 1-θ<ί<1 JO
where q*(t) = (i(l - i))^(log 1/(<(1 - t))f. Thus (12.2.27), (12.2.28) and
(12.2.8) imply that for any ε > 0
limlimsupP<| sup |Δη(2)|>ε>
= limlimsupPJ sup |Δη(1) - Δη(ί)| > ε} = 0. (12.2.40)
0-»O n->oo 4-0<t<l J
Similarly, (12.2.29), (12.2.30) and (12.2.8) imply that for any ε > 0
UmP{sup|A(i)|>e}
= lim Pi sup |Δ(1) - Δ(ί)| > ε} = 0. (12.2.41)
0-+° 4-0<t<i J
Hence Corollary 12.2.1 follows from Theorem 12.2.2 and Theorem 4.2 in
Billingsley (1968).
Proof of Theorem 12.2.3. Let θ and ε be as in (12.2.9). Since
θ > 1 + >/2, we can take r = oo, ν and ρ in Lemma 12.2.2 such that
2(0 + g) <y < 2Θ and υ<θ + 1< <e + 1+£u (12.2.42)
θ+ε-10-1 v J
Therefore, by (12.2.16) and (12.2.18), for any 0 < η < (ρ - 1 - θ)/θ, there
is а К < oo such that for any 0 < 5, t < 1
i=l
<K(np/2\t-s\p/v + n1+r>/2)
and
η
^ΐΣί7^ -f) ~ /(E7i -s) ~ с ~ s))| - Kn\l - si2/"'

330
Chapter 12 Empirical Processes
which imply that
E\an(t) - an(s)\* < K(\t - s\p/v + n-(p-2-n)/2)
and
E\an{t)-an{s)\2<K\t-s\2l\
Hence (Al) and (A2) hold for px = p/v > 1, P2 — ρ - 2 — η > 1, r\ = 0
and Г2 = 2/v. Note that 0<?7<(р—1 — #)/#, it is easy to see from
(12.2.42) that
. /Pi П +P2 r2\
mini —, , — I
V ρ p + p2 2 /
>i(l-i). (12.2.43)
By Theorem 12.1.2, (12.2.3) holds. This proves Theorem 12.2.3 by
Theorem 12.2.2.
Proof of Theorem 12.2.4. Prom Corollary 12.1.1 it follows that
(12.2.3) holds. By Theorem 1.1 of Shao (1995), we have for ρ > 2
c(>/2exp(c Σ p{2i)){E{I(U1 < t)
Е\£{1{Щ<1)-1{Щ<з)-{1-з))\
i=l
[log τ
<
~I(U1<s)-(t-s))2r/2
[log n]
+ nexp(tf Σ p2'lp{2i))E\I{U1 <t)
i=o
- m <s)-{t- β)\ή
[log n]
<c(np/2exp(c Σ Ρ(2ί))\* ~ S\P/2
[log n]
+ nexp(c£ Р2/р(2{))\Ь-з\).
Clearly, under condition (12.2.10), we have for ρ > 2 and any 0 < δ <

12.3 Strong approximations
331
min{e(p - l)/(2(2 + ε)), (ρ - 2)/2},
£?|Σ(/(^ < «) - ОД <s)~(t- «))
г=1
[log n]
<c(np/2\t-s\p/2 + nexp(c Σ р2/р(2{))\1-з\)
i=0
<c{npl2\t-s\pl2 + nl+8\t-s\),
since exp(cj^0 p2/p{21)) is a slowly varying function. This, in turns,
gives us for ρ > 2
£|αη(ί) - αη(β)|Ρ < c(\t - sf2 + n-(p-2-M)/2|t - s\)t
and for ρ = 2
#|an(t) - an(s)|2 < c\t - s\.
Thus (Al) and (A2) are satisfied for p\ — p/2, ρ2=ρ-2-2δ>0 and
П = r2 = 1. Hence by (12.2.6)
μ = (1+ρ-2- 25)/(p + ρ - 2 - 25) > 1/(2 + ε). (12.2.44)
On the other hand, under condition (12.2.11), we have for ρ > 2
E\J2(I(Ui<t)-I(Ul<s)-(t-sW
<c(np/2\t-s\pl2 + n\t-s\),
which implies that for ρ > 2
E\an(t) - an(s)\p < c(\t - s\p'2 + n'^^t - s\).
Thus (Al) and (A2) are satisfied for p\ = p/2, P2 — p — 2 and rx = Г2 = 1.
Obviously μ = 1/2. Now the proof of Theorem 12.2.4 is complete.
12.3 Strong approximations
In this section, we introduce the strong approximations for empirical
processes with a dependent sample. For simplicity, we only give the results
for an a-mixing sample due to Philipp (1982). Strittmatter (1990) has
given a further result for an absolutely regular sample.

332
Chapter 12 Empirical Processes
In order to discuss the strong approximations of empirical processes
with a dependent sample, we first introduce a result of strong
approximations for random elements with values in a Banach space. Let {xj,j > 1}
be a sequence of strictly stationary α-mixing random elements on
probability space (Ω,Σ,Ρ) and with values in a measurable space (A, A). Let
(S, || · ||) be a Banach space and let h : A —> S be a mapping. We call
{Xj := h(xj),j > 1} a sequence of strictly stationary α-mixing S-valued
random elements.
Theorem 12.3.1. Let {Xj,j > 1} be a sequence of strictly stationary
α-mixing S-valued random elements as above, such that \\X\\\ < ζ for some
ξ with Εξ2+δ < oo, 0 < δ < 1 and
a(n) < cn-(2+^1+2/*) (12.3.1)
for some ρ > 0. Suppose that for each m > 1 there is a linear mapping
Am : S —► S with the following properties:
sup ||Am|| < oo, (12.3.2)
m>l
dimAm5 < Cm, for some С > 0. (12.3.3)
And suppose that there is α θ with the following property: for each m > 1
there is an no(m) < cexp(ra1-^) and a non-increasing function g(m) >
m~1/2 such that for all η > no
P*{n~1/2\\ £(X,- - ЛтХ,)|| > д(т)} < πι'1'6, (12.3.4)
where P* is the outer measure of P. Moreover assume that for each m > 1
the mapping Лт о h is a measurable function from (A, A) into the linear
span LmS of Лт5, and that
EArnX1 = 0, £||AmXi||2+* < oo. (12.3.5)
Let Τ be the completion of the linear span o/Um>iAmS, so Τ is a separable
Banach space. Then there exists a sequence {Yj,j > 1} of i.i.d. Τ-valued
Gaussian random variables defined on (Ω,Σ,Ρ) such that EY\ — 0.
Moreover, for each s,t G T1, the space of bounded linear functional on T, the
following limits exist and the series converge absolutely and the covariance

12.3 Strong approximations
333
structure of Υχ is given by
Es(Y1)t(Y1)=JimoE{8(AmXi)t(AmXi)}
+ Jin^ Σ Е{8(АтХхЩАтХ^}
+ lim ^2Е{з(АтХ,)г(АтХх)} (12.3.6)
771—ЮО
i>2
and
ΙΙΣ№-^)ΙΙ<^
= o(n1/2{(togn)-<1-«/4
+ {\ognY-VlPg(a(\ogn)l-el2))) a.s.(12.3.7)
for some measurable Vn and some positive constant a and any β with 1 —
0/4</3< 1.
The proof of Theorem 12.3.1 will not be presented here.
Theorem 12.3.2. Let С С Л be a class of measurable sets with
N(6,C) < c6~r, for some τ > 1/6. (12.3.8)
Let {жп,п > 1} be a sequence of strictly stationary Α-valued a-mixing
random elements with
a(n) <cn~2r-4. (12.3.9)
Put gn(C) = I(xn G C) — P(C),C £ C. Then there exists a sequence
{Yj,j > 1} of i.i.d. Gaussian processes, defined on the same probability
space (Ω, Σ, P), indexed by С £ С with EYi(C) = 0,
E(Yx(C)Yx(D))
= Egx{C)gx{D) + £ Egx(C)gn(D)
n>2
+ £ Egn(C)gi(D), C.DeC (12.3.10)
n>2
such that with probability 1
suP|£№ieC)-P(<7)-is-(c))|
<Vn = 0(nx/2(logn)-A) (12.3.11)

334
Chapter 12 Empirical Processes
for some measurable Vn and some λ > 0.
The proof of Theorem 12.3.2 will need the following lemmas.
Let {ξη} be a sequence of strictly stationary a-mixing real random
variables with Εξι = 0, |£i| < 1 and a{n) < cn~p, where с > 0, ρ > 2.
We need an exponential bound for the partial sums of {&,i > 1}. Put
ρ = [nll2-% q = [n/(2pn)] + 1 for some 0 < к < 1/2. Define blocks
of consecutive integers Hj and 7j, 1 < j < q of length ρ each and Hq
consisting of η — 2p(q — 1) integers. Then Са,тд.Ня < 2p. The order of the
blocks is #ι, /χ, · · ·, Hq-i,Iq-i,Hq. We leave no gaps between blocks. Put
Уз= Σ &> zi = Σ&·
ieHj %eij
It is easy to see that
σ2 := Εξ\ + 2 £ £fc£n < oo.
n>2
Note that for sufficiently large η
ο ί 2σ2ρ, 1 < j < q,
Eti<\*J' Γα (12·3·12)
[5σ ρ, j =q.
Denote Cj = σ(τ/ι, · · ·, yj). We write
Уз =Yj+Vj, 1 < J <9,
where vj = E(yj\Cj-i),Yj = yj — £7(yj|£j_i). It is clear that (lj,£j,l <
j < g) is a martingale difference sequence and from Lemma 1.2.1, it follows
that
1Mb < 2\\yj\\00a1/2(p) < cp1""/2. (12.3.13)
Lemma 12.3.1. Let A > σ2. We have
p{E^(y/l£i-i) ^4Ата} ^ cA-v~p·
Proof. By the Holder inequality we have
E<y}\Cj-x) < E(^\Cj-i). (12.3.14)
By Lemma 1.2.1, we have
\Wvl\Cj-i) - Evj\\2 < 4||yi||200a1/2(p) < Ap2"»'2.

12.3 Strong approximations 335
Thus by the Minkowski and Chebyshev inequalities we have
р{£{Е{$\Са-Х)-Е$)>Ап}
< 16n-2A-2(qp2-^2)2 < cA-2p2-?.
Hence by (12.3.12) and (12.3.14), the probability in question does not
exceed
^{Σ E(y]\cJ-i) > Σ Η + An) ^ cA-y-f.
Lemma 12.3.1 is proved.
Lemma 12.3.2. For all R> 0 we have
Ρ{\ΈΥί\ > 5^1/2} < c{exp(-R2/A) + A'2p2'p}.
j<q
Proof. Let Μ be the index j of Hj or Ij containing n. Define
Σγ3> if* < Af,
Uk = { з<к
Um, \ik>M\
Υ^Ε{Υ2\ε^), iffc<M,
Mi
si = ( J<k
s2M, if к > Μ.
Obviously Uj - Uj-i = Yj < 4p =: C. If R < An^/p, we set λ =
Rl(AAnxl2), К = 20An, so that AC < 1. Put
Tk = exp(AE7fc - \x2{\ + \xC)s2k).
By Lemma 5.4.1 and Corollary 5.4.1 of Stout (1974) and Lemma 12.3.1 we
have
P\supUk >5i?n1/2}
<p{^Tk>e.V(x2K-\x2(l + \xc)s$}
< c{exp(-R2/A) + А~2р2-р}.

336
Chapter 12 Empirical Processes
But if R > Anxl2/p we set λ = l/(4p) and К = 20Rpn^2. Then AC = 1
and by the same calculation we obtain
P{supt/fc >5Дпх/2}
-20Rpn^2 + 3An\
16p2
< cA-2p2~f>.
^cexp(—Ϊ6Ρ—)+AV
Lemma 12.3.2 is proved.
Lemma 12.3.3. We have
Ρ{\Σνΐ\ * ™n1/2} < c(exp(-R2/A) + ηι+οκ-»'2(Α-2 + R~2)).
j<9
Proof. We have
j<9 j<9 3<9
By the Chebyshev and Minkowski inequalities we obtain from (12.3.13)
and expressing ρ and q in terms of η
Ρ{|Συ;| ^ Rnl'2} ^ cR-2n-l{qpl-pl2)2 < cR~2np~f.
j<9
Combining it with Lemma 12.3.2 implies Lemma 12.3.3.
Lemma 12.3.3 remains valid if we replace yj by zj,1 < j < q and
set zq = 0. Hence we have an exponential bound for the partial sums of
{€i,i > 1} as follows
ρ{\Σξ,<\>^ην2}
k<n
<c(exp(-i?2/^) + ^1+p/c~p/2(^~2 + ii~2))· (12.3.15)
Lemma 12.3.4. If the hypotheses of Theorem 12.3.2 are satisfied, then
for any given ε > 0, there exists α δ > 0, δ < се6 and no < cexpi — 1/(4ε))
such that for all η > no
P*{sup(K(C) - vn(D)\ :C,DG C, PX(C Δ D) < δ) > ε} < cexp(-^-),

12.3 Strong approximations
337
where
vn = nl'2{Pn - Px),
Π
pn(B) = n-1^2i(xjeB),
px(B) = p{xx ев), Be A.
Proof. Let r be so large that
2r > ε~6. (12.3.16)
Put 6k = 2~г~к(к = 0,1,2,···), mk = N{6k,C,Px), d{ = (i + l)~2e/32.
Take sets Ak\, · · ·, Akrn^ as in the definition of N(6k,C,Px), so that for
every С £ С and к = 0,1, · · · there exist r(k) = r(k, C) and s(k) = s(k, C)
such that Ащк) СС С Ащк) and Px(Aks(k)\Akr(k)) < 6k. Denote
Bk := Bk(C) = Aks(k)\Ak+hs(k+1), Dk := Dk(C) = Ak+1^k+1)\Aks(k).
Then Px(Bk) < 6k and Px(Dk) < 6k+1 < 6k.
Put no := ηο(ε) = ε2/(256<5ο). For every η > uq there is a unique
к = к(п) such that
1/2 < 86kn1/2/e < 1. (12.3.17)
Then for η > no, к = fc(n), δ = 6k and С £ С, г = r(k,C), s = s(k,C)
we have
vn(Akr) - ε/8 < vn(Akr) - δη1'2 < vn{C) < vn(Aks) + ε/8. (12.3.18)
From this we obtain
\vn{Aks(k)) - Vn(A0s(0))\
k-l
< ]T Wn(Ais(i)) - Vn(Ai+lAi+l))\
i=0
k-l
<Σ(Μβ·)Ι + ΜΑ)|). (12.3.19)
i=0
Let Bi be a collection of sets В = Ais\Ai+itt or Β = Α{+χ^\Α^3 with
PX(B) < δ{. Then for every С G С, Bi(C) and Д(С) G B{. The number
of sets in Bi
Card(Bi) <2m(i)m(i + l).

338
Chapter 12 Empirical Processes
Now let us estimate P{\vn(B)\ > di}, В G B{. Put ξη = I(xn G
B) - PX(B). Since
\Εξ!ξη\ < ||6||4||ίη||4(α(η - Ι))1/2 < Р^2(В)п-2-\
we have
oo
σ2 = Εξ2 + 2 £ Εξ1ξη < cPl'2{B).
п=2
Hence we can take δ/ such that σ2 < δ/ , it follows from (12.3.15) that
P{\MB)\ > di} < c(exp{-^6-1/2} + n—^^^(й-1 +г2)).
Then from (12.3.8) and taking к = l/(8r + 16) we have
Pi := P{\vn(B)\ > di for some В € Bi}
< сехр((-в2/(6г1/2196 · 322(г + Ι)4)) · δ~2τ + n'^X1-2*).
Therefore by (12.3.16) and (12.3.17) and taking r large enough, we obtain
E^<c(^p(-^)+«~1/4^(1+2t))
<cexp(-l)
for η > щ. Put Qn = P(Vn > ε/8), where
Vn = s\ip{\vn(Akr) - Vn(Aks)\ : Akr С Aks,
P(Aks\Akr) <6k, r, 5 = 1, · · ·, mk}.
Then from (12.3.15) and (12.3.17) it follows that for η > n0
Qn < c8^ exp(^ · δ?*) + „—3/V~2T
<cexp(-l).
By using (12.3.16) again we have
po := P{sup(|i/n(Aoi) - 1/„(Ау)| : P(A0iAA0j) < 3<50) > ε/4}
< с^2техр(-^о"1/2) +n-1-T+P4"1-2T
<cexp(-i).

12.3 Strong approximations
339
Lemma 12.3.4 is proved.
Proof of Theorem 12.3.2.
Let S be the space of all bounded real-valued functions on C. If / G S
we set ll/ll = s\ipCeC \f(C)\. If χ G Л we set h(x) = I(x G C)-P(C). Thus
h : A —> S. Let m > 1 and put ε = m-1/(6r). Find δ and no according
to Lemma 12.3.4. Then δ < с m~llT and n0 < cexp^m1/^/Δ. Let
Ai,···,^, d = Ν(δ), be sets such that for all С G С there exists Лг
with P(Ar AC) < δ and r minimal. Now Ν(δ) < οδ~τ < cm. We define
Am : 5 -> 5 by АтЛ(ж) = /(ж G Д.) - P(Ar) if Л(ж) = /(ж е С) - P(C)
with Р(Д. Δ С) < δ. Then dimAm5 < cm. From Lemma 12.3.4 (with
D = Ar) we conclude that (12.3.4) is satisfied with g{m) = m~1^6r^ and
θ = 1 — 1/(6τ). The result follows now from Theorem 12.3.1 choosing β
close to 1, but subject to 1 — θ/4 < β < 1. Theorem 12.3.2 is proved.
From Theorem 12.3.2 we have the following theorem immediately.
Theorem 12.3.3. Let {Xj,j > 1} be a sequence of strictly stationary
α-mixing a-dimensional random vectors with a distribution F(x) and
a(n) = 0(n-4-2d). (12.3.20)
Denote gn(s) = I(Xn <s)- F(s), for s G Rd. So the series below defining
the covariance function
oo
Γ(8,θ') = Egi(s)gi(s') + Σ E(9l(s)gn(s') + gn(s)gi(s'))
71=2
converges absolutely for 5,5х G Rd. Then there exists a sequence {lj,j >
1} of i.i.d. Gaussian processes, defined on the same probability space
(Ω,Σ,Ρ), indexed by s G Rd with
EYx(s) = 0, EYlWY^s') = Г(5,5;), 5,5х G Rd
and a positive constant λ depending on d only such that with probability 1
sup |Σ(/(Χ,- < s) - F(s) - ВД)| = 0{nll\\ogn)-x).
seRdj=i '
Proof. Let Ρ be the probability measure induced by F. Let Fi(si),
1 < г < d be the г-th marginal of F(s), s = (s1? · · · ,5^). Let r > 1 be
given. We define
eij = mvFi(j2~r), 1 < г < d, 0 < j < 2r.

340
Chapter 12 Empirical Processes
Let С be the collection of all intervals С = (—00,5], s G Rd. For any
С G С there exist Ap and Aq both of the form (—00, (si^, · · · ,s<#d)] for
some (sijx, · · ·, Sdjd) such that
ApcC cAq and Р(ДДЛр) < d2~r.
The collection of all such sets Ap has cardinality < 2dr, i.e. -/V(d2~r,C, P)
< 2dr. Hence by interpolation (12.3.8) is satisfied with τ — d and с = (2d)d
and (12.3.10) holds because of (12.3.20).
Remark 12.3.1. Philipp and Pinzur (1980) gave an almost sure
approximation of the multivariate empirical process by a Kiefer process. Let
{Xn,n > 1} be a strictly stationary α-mixing sequence of random vectors
in Rq with continuous distribution F and
a{n) = 0(η-4-*(1+ε))
for some 0 < ε < 1/4. The empirical process of {Χη,η > 1} is defined as
R(s, t) = [t](F[t](s) - F(e)), t > 0, a G Rq.
Let
00
Γ(β, s') = E{9l(s)gi(s')} + Σ E{gi(s)gn(s') + gn(s)gi(s')}, s, s' € Rq
71=2
where gn(s) = I(Xn < S)~F(S)· Then without changing its distribution we
can redefine the empirical process {i?(s, t), s G i?9, t > 0} on a richer
probability space on which there exists a Kiefer process {K(s, £), 5 G i?9, t > 0}
with covariance function (t Л ^)Г(5,5Х) and a constant λ = A(g, ε) such
that
sup sup \R(s,t)-K(s,t)\ = 0(T1/2(logT)-x) a.s. (12.3.21)
t<T s£№
When q = 1 and {Xn5 η > 1} are uniformly distributed over [0,1], (12.3.21)
coincides with the result by Yoshihara (1979), but with less mixing rate.
By Theorem 1.15.1 in Csorgo and Revesz (1981), (12.3.21) implies that
the Strassen-type law of iterated logarithm holds true for
{%(«) = i?(5,t)/v/2Uoglogt,0 < s < 1}.

12.4 Moduli of continuity of empirical processes
341
12.4 Moduli of continuity of empirical processes
Let {Xn, η > 1} be a sequence of random variables with a common
distribution F(x), {/3п(0э ~~°° < ^ < oo},n = 1,2,···, a sequence of its
empirical processes. The moduli of continuity of empirical processes are
defined as follows:
wn(an) = sup \Pn(t) - Pn(s)\.
\t—s\<an
— oo<s<£<oo
Stute (1982) proved for the i.i.d. case the following theorem.
Theorem 12.4.1. Suppose that
(i) 0 < an < 1, an [ and nan | oo,
(ii) nan/ log η —► oo,
(Hi) log a~x I log log η —► 0.
Then
lim (anloga~1)~1/2^n(an) = 1 a.s. (12.4.1)
η—юо
Definition 12.4.1. Let 0 < λ < 1, А С R. The function g(x) is said
to satisfy the uniformly local λ-order Lipschtz (λ-ulL) condition on Л, if
there exist δ > 0, Μ < oo such that
sup \g(x + z)- g(x)\ < Μ\ζ\χ, \ζ\ < δ. (12.4.2)
xeA
Zhou (1994) discussed the moduli of continuity of empirical processes
when the sample is (^-mixing or α-mixing, and proved the following
theorems.
Theorem 12.4.2. Let {Xn,n > 1} be a sequence of strictly stationary
φ-mixing random variables with a common distribution F(x). Suppose
that F(x) satisfies the 1-ulL condition and Υ^ι φ1Ι2{2%) < oo. If there is
a sequence of positive integers {mn} such that
1 < ^n < n, (p(mn) < A and ( ) mn < C,
rnn V nan /
where A and С are two constants, then
wn(an) = 0((an log n)1/2) a.s. (12.4.3)

342
Chapter 12 Empirical Processes
Theorem 12.4.3. Let {Xn,n > 1} be a sequence of strictly stationary
α-mixing random variables with a common distribution F{x). Suppose that
F(x) satisfies the 1-ulL condition and a(n) = 0(pn) for some 0 < ρ < 1
and an —> 0(n —► oo). Then for any 0 < θ < 1 we have
wn(an) = θ[αη2 log2 nj a.s. (12.4.4)
Remark 12.4.1. If F(x) satisfies the λ-ulL condition (0 < λ < 1)
on A(c ii), then (12.4.3) and (12.4.4) are rewritten as
wn(an, A) := sup \βη{ι)-βη(8)\ = 0((a* logn)1'2) a.s. (12.4.5)
t,seA,\t—s|<an
and
wn(an,A) = θ(αη2 log2 nj a.s. (12.4.6)
Remark 12.4.2. Because of Lemma 12.4.1 below, the condition
Ση^ι ψ(η) < °°5 which is required in Zhou (1994), is weakened to Y^L^
•(^1/2(2i) < oo in Theorem 12.4.2.
The proof of Theorems will need the following Bernstein type
inequalities. '
Lemma 12.4.1. Let {Xn,n > 1} be a φ-mixing sequence with EXn =
0, \Xn\ < d, EX* < D and ΣΖιΨ1/2(2ί) < °°· Then there is ci =
@ι(φ(')) > 0 such that
P{\J2Xi\>e}
< expJ3>/en^^- - as + Cia2Dn\
where a is a real number, m is a positive integer satisfying m < η,α < η
and amd < 1/4.
Lemma 12.4.1 is an improvement of Lemma 1 of Collomb (1984). Its
proof follows Collomb's lines provided one uses Lemma 2.2.2 instead of
Lemma 1.2.10 which is employed by Collomb (cf. Lemma 11.1.1).
Lemma 12.4.2. (Doukhan, Leon and Portal 1984) Let {Xi,i > 1}
be an α-mixing sequence with EX{ = 0, \X{\ < 1 and a{n) < Cpn. Denote

12.4 Moduli of continuity of empirical processes
343
σ — sup||X;||7, where 7 = 2/(1 - 0),O < θ < 1. Then there exist Οι,02
which depend only on α(·) such that
where
σ
\C2 if η1/2σ < 1,
r1/2 if η1/2σ>1.
Proof of Theorem 12.4.2.
As we have mentioned in the beginning of this chapter, we need only
to consider uniform empirical processes. And by the 1-ulL condition, it is
enough to show
sup sup |αη(ί) - otn{s)\ = 0((an logn)1'2) a.s. (12.4.7)
0<M<1 \t-s\<Man
where αη(·) is a uniform empirical process. Without loss of generality, we
can assume that Μ = 1 in (12.4.2).
Divide the interval [0,1] into Kn subintervals by points to = 0? tj =
j/Kn, j = 1, · · ·, Kn, where Kn = [a~x logn]. Denote
Vn(t) = sup \an(t) - an(s)\.
\t-s\<an
For any fixed t,s G [0,1] with \t — s\ < an, there are two cases:
(i) 5 and t both fall into the same subinterval, i.e. there is a j, 0 <
j < Kn — 1 such that s,t G [^,^+ι]. Then
\oin{t) ~ OLn{s)\ < \an(t) - an(tj)\ + \an(tj) - an(s)\
<2nmax Vn(tj).
(ii) 5 and t fall into the different subintervals, i.e. there exist j and
r, l<j + l<r< Kn such that 5 G fo^j+i], t G [2r,2r+i]. Since
\s — t\ < an, tr — tj+1 < an. Then
\oin(t) -an(s)\
< \<xn(t) - OLn{tr)\ + |an(<r) - an(<j+i)| + |an(<j+i) - an(s)\
0<j<Kn
Hence, in any case, we have
sup \an(t) -an(s)\ < 3 max Vn(tj). (12.4.8)
\t-s\<an l<J<Kn

344
Chapter 12 Empirical Processes
For any fixed j, 1 < j < Kn, divide the interval [tj — an,tj + an] by
points
Vjr = ^j "ι ^"Τ j Τ = un, 0n + 1, · · · , un 1, un,
"η
where bn = B[(nan/logn)1/2], constant В will be specified later on.
Denote <t)jr = n~1/2\an{tj) — an{r]jr)\. For any given 5 G [tj — an,tj + an],
there is an r, —bn < r < bn such that 5 G ferj^r+i]· By monotoncity of
the empirical distribution En(t), we have
n~1/2Vn(tj) = sup |(Sn(ij) - b) - (Sn(e) - 5)|
\tj— s|<an
- Ла^ Р2* \(En(tj)-tj)-(En(s)-s)\
-bn<r<bn Vjr<s<Vj,r+l
- um^u il^Ci) " *i " (En(Vjr) ~ 4j,r+i)|,
-Οη<Τ·<6η
l^n(ij) - «j ~ (^nfer+l) - 4j,r+l)|}
- г^Ъг. ί^>' <^> + l} + \Vjtr + l - Vjr\
-bn<r<bn
< max {(f)jr} +an/bn. (12.4.9)
-bn<r<bn
Write
-. η η
Фзг = |- Σ[Ι{4τ <Ui< tj) - (tj - Vjr)}\ =: \Σ Zi[ (12.4.10)
m . . .
г=1 г=1
Obviously |Z;| < 2/ra, £Z; = 0 and £Z? < an/n2. Take ε = B(an logn/n)1/2
and a = (B~1na~1 logn)1/2. By the assumption of the theorem, for large
β we have
/nlogn\i/2 2 0/logn \ 1/2 1
amnd= — I mn- = 2 — 1 mn<-.
V £?αη / ?г \Впап' 4
By Lemma 12.4.1 we have
ρ{|ί>|>4
i=l
< C\ expj —αε + Cia2an/n >
< Cx expj-Б1/2 logn(l - Ci/S3/2)}. (12.4.11)
where C\ — exp(3y/eA). Choosing В large enough, we have
P\ max max |<Д»>| > i?(an log n/n)1'2 \
^ 0<j<Kn -bn<r<bn >
< cKnbnn-Bl'2l2 < en'2.

12.4 Moduli of continuity of empirical processes
345
Therefore, from the Borel-Cantelli lemma it follows that
max max |<Д->| < B{an\ogn/n)1'2 a.s. (12.4.12)
0<7<^n -bn<r<bn
Combining it with (12.4.8) and (12.4.9) yields (12.4.7). Theorem 12.4.2 is
proved.
Proof of Theorem 12.4.3.
The proof is along the same lines of that of Theorem 12.4.2 with Lemma
12.4.2 instead of Lemma 12.4.1. Let 0 < θ < 1, 7 = 2/(1 - θ). It is
clear that Ε\Ζ{\Ί < 2η~7αη, г = 1,···,η. Hence σ := sup{||Z;||7 : г =
1,···,η} < 21^n-1a1nh and ηιΙ2σ < 2lhn-1l2a1J1 < 1. By Lemma
12.4.2 with ε = εη = В(п'^2с^'е)/21о^п), we have
Р{Фэг > εη} < С1ехр{-С2В(п'1а1гГвlog4п)г^/п'г^а}/^}
< Ciexp{-C2Blogn}.
Therefore for large £?, we have
P\ max max ώ7> > εη} < cn~2.
l0<j<Kn-bn<r<bn^J

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Chapter 13 Convergence of Some Statistics
with a Mixing Sample
Large sample theory in statistics is an important subject. In general,
the sample is assumed to be independent. But, in some practical cases, the
observations are dependent. In this part, we shall give some large sample
properties for several interesting and useful statistics, such as U-statistics,
error variance estimations in linear models, density function estimations
with a mixing sample.
13.1 [/-statistics
Let {Xn,n > 1} be a strictly stationary sequence with a common
distribution F(·), h : Rm —> R be a symmetric function in its m arguments.
A [/-statistic is given by
U"=(n) Σ ^n-^J n>m.
Here h is called a kernel function of Un. This class of statistics was
introduced by HoefFding (1948) as a generalization of the sample mean. Many
statistics of interest fall within this class or may be approximated by a
member of this class.
Remark 13.1.1. [/-statistics are closely connected with another class
of statistics, the so-called von-Mises statistics (von Mises 1947) defined by
η η
*1=1 *m = l
These two kinds of statistics have similar limit behavior. So we discuss
only [/-statistic as a representative.

348
Chapter 13 Convergence of Some Statistics with a Mixing Sample
A kernel h is called degenerate (for the distribution F) if for all choices
of a^, 1 < г < m and every J G {1, · · ·, ra},
Eh{ai, · · · , α^-ι,Χ^,α^+ι, · · · , am) = 0;
a {7-statistic will be called degenerate if the corresponding kernel has this
property.
First of all, we introduce an important tool in deriving the asymptotic
theory of {/-statistics, HoefFding's projection method. Put
θ= I'- I h(xu , xm) Π dF(xk), (13.1.1)
hr(xi,---,xr)= ··· h(xi,---,xm) Π dF(xi),
i=r+l
r = 1, · · · ,ra — 1,
and
hr(x\, · · · , #r) = hr(xi, · · · , #r) — 0, r = 1, · · · , ra — 1.
The projection of Un is defined as
η
m
un = -Y/h1(xi) + e.
г=1
ίΛι — Un may itself be expressed as a {/-statistic
V / l<tl<-<*m<n
=: Дп, (13.1.2)
where
H(xu· · · ,#m) = /i(xi,- · · ,#m) - Λχ(χι) - · · · - hi(xm) - θ
is a degenerate kernel. We call Rn the remainder of Un.
At first, modifying definition of a (^-mixing sequence, we call a
sequence {Xn,n > 1} (£*-mixing or (^-mixing in both directions of time if
the sequence itself and the time reversed sequence are (^-mixing, that is
φ*(η):=8\ιρ sup mzx{\P(B\A) - P(A)\, \P(B\A) - P(B)\} -+ 0
B^k+n
as η —> oo.

13.1 U-Statistics
349
Obviously, (£*-mixing implies (^-mixing.
In this section, we shall establish weak and strong convergence for a
<£*-mixing sequence.
Denker and Keller (1983) proved the CLT and their rate of convergence,
functional CLT and a.s. approximation by a Wiener process. Combining
the results of weak convergence and strong approximation for a (^-mixing
sequence in Chapters 5 and 9, we can weaken the conditions on moments
and/or φ*(ή).
13.1.1 Bounds for the remainder Rn
Let {Χη,η > 1} be a strictly stationary <£*-mixing sequence. Assume
s2 := sup E(h(Xtl, · · ·, XtJ)2 < oo. (13.1.3)
1<*1<."<*т
First, we cite two lemmas given by Denker and Keller (1983). Lemma
13.1.1 is a conditional version of Lemma 1.2.8 with ρ = q = 2. Let
Л, #, B\, B2 be sub-a-fields of T. For probabilities Ρ and Q on Τ define
the distance of Ρ and Q over Λ given В by
d(P, Q : A\B) = sup \P(A\B) - Q(A\B)\.
АеЛ,вев
Moreover put
d(P : A\B) = sup \P(A\B) - P(A)\.
Аел,вев
Lemma 13.1.1. Let /i and /2 be an AM B\- and A V B^-measurable
function, and let P, Q\ and Q2 be probability measures coinciding on A.
Then
\EP(hf2) - EP[EQl{h\A) · EQ2(f2\A)}\
< (4 + 2V2)
•тах{^/2(Р : Вг\А V B2),d1/2(P, Qx : B1\A),d1/2(P,Q2 : B2\A)}
.{^(/^)1/^^1(/nl/2}·max{^(/|)1/^JE;Q2(/|)l/2}.
Lemma 13.1.2. Let f be an Αν Β-measurable function, and let Pn(n >
1) and Q be probability measures coinciding on A. If
lim d(Pn,Q:B\A) = 0,
Π—ΚΧ)

350
Chapter 13 Convergence of Some Statistics with a Mixing Sample
then
EQ\f\<liminfEPn\f\.
Lemma 13.1.3. For given ε > 0, there exists а С = C£ > 0 such that
ER2n < Cn-2+£s2 (n>m).
Proof. By (13.1.2), it suffices to estimate the variance of a degenerate
{7-statistics. We shall show: if h is degenerate, then
(Л EU2 < cn^-^s2. (13.1.4)
For a = (ai,---,am),b = (bi,---,bm) £ Nm, we put
W(a,b) = X)fe(Xtl,---,Xtm), (13.1.5)
where the summation extends over all indices £i,-",£m satisfying щ <
U < bi and U Φ tj for 1 < i φ j < m. Putting 1 = (1, · · ·, 1) G Nm, we
thus have
(n)Un = —W(l,nl). (13.1.6)
\rnl ml
For the estimation of £'(W(l,nl))2 we proceed recursively, decomposing
W(l, nl) into sums over smaller index-blocks (This is inspired by the proof
of Theorem 2.1.2).
We need some preparations: let (Ji )n>i5 * * *, (Xn )n>i be m
independent copies of the sequence (Xn)n>i5 and put for q G {1, · · ·, m}m
where the summation extends over the same index-set as in (13.1.5). Let
In = {(a, b) G N2m : bi = щ + η — 1, αι = aj or |a; — a,j\ > η for all г, j}
and define
т(п) = sup{£;(W(a, b; q)f : (a, b) G In, q G {1, · · ·, m}™}.
Consider fixed non-negative integers A:,/,p, η with η = kl + p. For each
(a, b) G In and q G {1, · · ·, ra}m we have by the Holder inequality and the
triangle inequality:
|£(W(a, b; q)f - E(W(a, b - pi; 9))2|
< (r(n)1/2 + A;mr(/)1/2)mpnm-1s, (13.1.7)

13.1 U-Statistics
351
where we have tacitly assumed that
sup E[h(x[?\. · ·,X^))2 < s2 (13.1.8)
l<il<-<tm
for all q G {1, · · ·, ra}m, what can easily be proved using repeatedly Lemma
13.1.2.
Now decompose W(a, b — pi; q)2 as
Ща, b - pi; qf = ]T W(& + hi, a + l(u + 1) - 1; q)
u,ve{0,-,k-l}™
.W(a + lv,a + l(v + l)-l;9).
Each Г(и) := W(a + /u,a + /(u + 1) — l;g) is determined by m blocks
of time coordinates of length / (possibly counting a block several times).
Denote for a fixed pair (u, v) these 2m blocks by i?i, · · ·, i?2m and assume
inf Bi < inf B{+i(i = 1, · · · ,2ra — 1). With this convention it is not hard
to see that
Card{(u, v) : sup Si + / > inf B2 and inf i?2m — I < supi?2m-i}
< Cmk2™-2, (13.1.9)
where Cm is a combinatorial constant depending only on m. For all pairs
(u, v) not belonging to the set described in (13.1.9) we can apply Lemma
13.1.1 in the following way: assume that supi?i + / < inf B<i and that B\
is a block stemming from Г(и). (The remaining three cases are treated
exactly in the same way.)
Choose /1 = Г(и),/2 = Γ(ν),Βι = a(Xut G B1),B2 trivial and
A = a(Xt,teB1U---UB2m).
Observing (13.1.8) we then get
|£Γ(ιι)Γ(ν)| < (4 + 2л/2)^*(01/2т(0,
because h is degenerate. It is for this application of Lemma 13.1.1 that we
have to introduce independent copies of the original process, and since in
two of above mentioned four cases, we have to single out the block 52m
(instead of Βχ), we need (^-mixing in both directions of time. Combining
the last estimate with (13.1.7) and (13.1.9) we obtain
W(a, b; q)2 < Стк27П-2т(К) + k2m(4 + 2Λ/2)<^*(/ι)1/2τ(0
+ Ып)1/2 + kTnr(l)1/2)mpnTn-1s

352
Chapter 13 Convergence of Some Statistics with a Mixing Sample
and taking the supremum over (a, b) G In and q G {1, · · ·, m}m
r(n) < k27n-2r{l){Cm + (4 + 2V2)*V(01/2)
+ {Tin)1'2 + k7nr{l)1l2)mpn7n-ls. (13.1.10)
Take к to be large enough such that (Cm + (4 + 2\/2))k~£ < 1/4. Then
choose no = min{s : A:2(^*(5)1/2 < l}. For / > no and ρ < к, (13.1.10)
implies
(r(n)i/2 _ 1тоыт-г5)2 < (-fcm-1+£/2r(01/2 + ^"^mb"1-^)2,
and hence
r{n)ll2 < -кт-1+£12т{1)1'2 + (- + kl-*l2)mknm-ls. (13.1.11)
Given η choose lo,h,· · · ,lr such that /о = Щ^г-1 = &ii + Pi for some
0 < Рг < k(i = 1, · · · ,r) and no < lr < kriQ. Apply (13.1.11) to each pair
(li-i^h) to obtain
r(ii-i)1/2 < \кт-1+^2т(1^2+А1^1\
where A=(—l· kl~e'2)mk. By induction this leads to
τ{ηγ'2 < 2-Tk^m~1+e^r(lT)1/2 + Anm-1+£/2s £ 2_i
3=0
< nm~1+e/2(——!^— + 2As)
< nTO-1+e/2((fcno)1_e/2 + 2A)s,
since η > Α;Γί,. and /Γ < kno. If we put С = ((кщ)1-*/2 + 2A)2(m\)~2, we
get
т(п) < Cs2n2m~2+£(m\)2 (13.1.12)
. and (13.1.4) follows from (13.1.6).
Lemma 13.1.4. Assume that condition (13.1.3) is satisfied. Then for
any ε > 0 and c^ > 0 we have
R^ = 0(η-3/4+ε) a.s.

13.1 U-Statistics
353
and
P{ max n\Rn\ > cN\ = OtJV1/2^2).
Proof. By (13.1.2) it clearly suffices to prove the lemma for a
degenerate {7-statistic with kernel h. Put
Z(p, q) = W(l, (p + q)l) - W(l,pl) (p, q G N).
If 2r~1 < η <2r and η = Σ£=1 ά{2τ~% denotes the dyadic expansion of n,
r k—1
W(l,nl) = Z(0,n) = Σ z(X)di2r-i,djb2r-fc).
fc=l t=l
For г, гх G N, / = 1, · · ·, r and j = 1, · · ·, 2/ consider the sets
ETj:r = {\Z((j-l)2r-l,2r~l)\>ar,u},
where Οίγ ιι are constants to be chosen later. We shall show below that
E(Z(p,q))2 = 0(q(p + q)b~l), b = 2m - 3/2 + ε. (13.1.13)
By the Chebyshev inequality P{E$) = 0(a~pf<r-t)jb-1).
The a.s. bound for Rn follows now from the Borel-Cantelli lemma,
because for агД = 2b(r-1)/2(r - l)3/r we have
P\ max |Z(0,n)| > n6/2(logn)3)
2"— !<n<2
r 2'
<EEp(*#)+°(r-3).
1=1i=i
To prove the maximal inequality, let R,N be given such that 2R~l <
N < 2R. Putting ar,N = 2^-1^7η~1>~1 cN for r < R, it follows that
P{ max n'm+1\Z%n)\> cN\
I Kn<N ι ν J /ι - j
d or 1
pfU U {^(ο,η)!^*»-1**}}
1<η<Ν
R 2r-l
< _
r=ln=2r~1
ζ ΣΣ,Σ,ρφ") = ociv^iogAoV)·
r=l/=1j=l
We still have to prove (13.1.13). If q > p, it follows from (13.1.12) that
E(Z(p,q))2 < {(ΕΖ(0,ρ)ψ2 + (ΕΖ(0,Ρ + ς)ψ2}2
= 0((P + q)2m~2+*) = 0(q(p + q)b~3/2).

354
Chapter 13 Convergence of Some Statistics with a Mixing Sample
If Ρ > Я > Ρ1/2 we obtain similarly
E(Z(p + q)f = 0((p + q)2™- 2+e) = 0(q(p + q)^1).
Now consider the case of q < p1!2. Put In = {(a, b) G N2m : 6; =
di+ri— 1 for exact ra—1 coordinates and a^ = £); else; a{ = ay or \ai—aj\ >
η for all г, j} and define
r(n) = sup{£(W(a, b; <?))2 : (a, b) G /„, <? G {1, · · ·, m}m}.
Then similarly to (13.1.12), we can show
r{n) < Cn2m~3+£s2
for some С > 0. Hence we obtain
E(Z(p,q))2 = E(j2z(P + k>1))
k=0
<
= 0{q\p + qYm~^) = 0(q(p + qf-1).
This finishes the proof of Lemma 13.1.4.
13.1.2 WIPforUn
Let
a2n = E(Y,h1(Xi))\
i=l
nt
Wn(t) = (U[nt]-e), 0<ί<1.
nm re l J
πιση
Theorem 13.1.1. Let h be a non-degenerate kernel. Assume that
condition (13.1.3) is satisfied and σ2 —> oo. Moreover, assume that for any
£ > 0
lim 4#^i№)2/(M*i)l > ε*η) = 0.
η—>oo /τ*
η
T/ien Wn => Wasn-^ oo.
Proof. Put
wn(t) = —(u[nt]-e), o<t<i.

13.1 U-Statistics
355
By Corollary 5.1.4, Wn => W as η —> oo. Theorem 2.1.2 implies that
σ2/η1-ε —> oo as η —> oo for any ε > 0. Hence by the maximal inequality
of Lemma 13.1.4 we obtain
P{ sup -HL\R[nt]\>£\
1 o<t<i rnan L J >
< P{ sup ni|i2fntl| > smn(1~e)/2}
= OirC1'2*2*).
Hence the result follows from (13.1.2).
Remark 13.1.1. Denker and Keller (1983) and Zhang (1989)
discussed the Berry-Esseen inequality for {7-statistics with a (£*-mixing
sample. We write the Zhang's result without any proof:
Let {Xn,n > 1} be a strictly stationary <£*-mixing sequence, Un be a
{/-statistic with kernel h{x\,X2). Denote
oo
σ2 = ЕНЦХг) - θ2 + 2 £{£M*i)M**+i) - θ2}.
k=l
Suppose that there exist constants С > 0, β > 0 such that
φ*(η) < Οβ~βη.
If σ2 > 0 and
sup E\h(Xi,Xj)\3 < oo,
l<i<j<n
then for any ε > 0 we have
.... υη-θ
sup
pi U" θ <χ\-φ(χ)
< en
-1/2+e
13.1.3 Strong approximation by a Wiener process for Un
Theorem 13.1.2. Let h be а поп-degenerate kernel. Assume that
sup E\h(Xtl,-~,Xtm)\2+6 <oo for some δ > 0. (13.1.14)
l<ti<-<tm
Moreover, assume that
(i) &n ^ cin for some c\ > 0,
(ii) <p*(n) < C2n~a for some c<i > 0 and a > 0.

356
Chapter 13 Convergence of Some Statistics with a Mixing Sample
Then one can redefine {Xn,n > 1} without changing its distribution on a
richer probability space together with a Wiener process W(-) such that
^(Un -Θ)- W{al) = 0(σ2/(2+δ)(1ο§ση)1+ε+(1+λ)/(2+5)) β...
for any ε > 0, where λ = 2(log3)/logr-1 and τ = 1 — 2(α — 1)/α(2 + δ).
Proof. By Lemma 13.1.2, condition (13.1.14) implies that £|/h(Xi)|2+* <
oo. Therefore from Remark 9.1.1, we conclude that we can redefine the
sequence {Λι(Χη), η > 1} on a new probability space together with a Wiener
process W(-) such that
J2~hMi) - W{al) = 0(σ2/(2+δ>(1ο6ση)1+ε+(1+λ>/(2+*)) a.s.
г=1
In fact, it is not hard to see that on the new probabilty space we also can
redefine {Xn} itself, for example by considering strong invariance principles
for i?2-valued random vectors. By Lemma 13.1.4, nRn = 0(n1/4"l_e) a.s.,
hence the theorem is proved.
13.1.4 SLLNforUn
Wang (1994) proved a SLLN for {7-statistics with a <£*-mixing sample.
We consider only the case of m = 2. In order to prove the theorem, We
need the following lemma, which was proved in the proof of Theorem 3 in
Babbel (1989):
Lemma 13.1.5. Let h be a degenerate kernel. Suppose that condition
(13.1.3) is satisfied and
φ*(η) = 0(n~(4+<5)) for some δ > 0. (13.1.15)
Then
e( max υλ < cn~2s2.
\l<i<n /
Theorem 13.1.3. Assume that condition (13.1.15) is satisfied and
s\ipE\h(XuXn)\ < oo. (13.1.16)
n>2
Then we have
Un —► θ a.s. as η —> oo.
where θ = f f h(xi, X2)dF{xi)dF{x2).

13.1 U-Statistics
357
Proof. For к € Ν, put
hW(xux2) = h(x1,x2)I(\h(xl,x2)\ < 22k),
*(*) = J J h^(x1,x2)dF(x1)dF(x2),
h[k\x) = Jh(k\x,y)dF(y),
Н^(хих2) = h^k\xux2)-hf\xl)-hf\x2) + e^
and
\ ) l<i<j<n
2 n
τι л
*{пк) = (пХ Σ ^\xhXj).
l<i<j<n
The last one is a {7-statistic with a degenerate kernel H^k\ It is easy to
see that
and hence we can write
limsup|i7n — θ\
η—>oo
< lim sup max {IE/<*> - 20^ Ι + Ι Δ<*> Ι
fc^oo 2*<η<2*+!
+ |[/n-[/,(fc>| + |0-0(fc>|}. (13.1.17)
Obviously, condition (13.1.16) implies that
0_0(*)_>O asA:->oo. (13.1.18)

358 Chapter 13 Convergence of Some Statistics with a Mixing Sample
Applying Lemma 13.1.5 and condition (13.1.16), we have for any ε > 0
oo
УР{ max |Δ^|>ε|
fc=l —
oo
< C£-2^2-2fcsup^/i2(X1,Xn)7(|/l(X1,XTl)| < 22k)
k=x n^2
< ce~2 sup £ 2~2k Σ Eh2(Xu Xn)I{22^ < \h(XuXn)\ < 22>)
n^2k=i j=x
OO
-2k
< се-2Б^^Ек2{ХъХп)1{22^-^ < \h(XuXn)\ < 22^')^2
< ce~2supE\h(XuXn)\ < oo,
n>2k=l k=j
n>2
which implies
limsup max ΙΔ^Ι = 0 a.s. (13.1.19)
Moreover
£p{ U {UntuX*}}
к=1 2к<п<2к+1
oo
<Σ^{ U (U тъ,*)]>*"·})}
к=1 2к<п<2к+1 l<i<j<n
oo
< £22<*+1>supP{|/i(*i,*„)| > 22k}
fc=i n^2
OO OO
< 4sup Σ 22* Σ P{22' < |Л(ХЬ Х„)| < 22^+1)}
«>2fc=l j=k
<
0 3ΜΡΣΕ\ΗΧι,Χη)\Ι(2ν < |Л(*ь*п)| < 22^+χ))
n>2j=1
< csupE\h(Xi,Xn)\ < oo,
n>2
which implies
limsup max \Un - UJ.k)\ = 0 a.s. (13.1.20)
Estimate the first term of the right hand side of (13.1.17). Put
h[k\x) = h{k)(x)I(\h[k)(x)\ < 2k)

13.1 U-Statistics
359
and write
2 Л
;(*)/
|i/w _ 20(*)| < |^ Σ№ }TO - ^i №))
i=l
(*)/
+ |^Σ№)№)-ΛΓ)№))
i=l
+ 2\θ^ - Eh^iXi)]. (13.1.21)
Consider \θ^ - Eh^iX^l first. We have
|0(*) _ ^(xfc)(Xx)| = lEhPiX^Idh^iX!)] > 2fc)| -► 0 (13.1.22)
as к —> oo. Similarly to (13.1.20), we have
oo η τι
Σ^{ U {E4fc)№)^E^fc)№)}}<^
k=l 2k<n<2k+1 i=l
г=1
which implies
■ΐτ*£8«Ι; £$"<*>-Я4**»
= 0 a.s.
(13.1.23)
Furthermore, {Л^р^-ЯЛ^рСДп > 1} is a strictly stationary and
<£*-mixing sequence with mixing coefficients φ*(η) = 0(n~^4^). Hence
using Lemma 2.2.10 we obtain
oo 9 n
fc=l
г=1
uu
<c^2-^(^(X1))2
k=l
oo
<c^W№)l<~.
fc=l
which implies
limsup max - J^(h\k)(Xi) - ΕΐΨ\χΛ) = 0 a.s.
i=l
Combining (13.1.22)-(13.1.24) with (13.1.21) yields
limsup max |t/<*> - 20<*>| = О a.s.
(13.1.24)
(13.1.25)

360
Chapter 13 Convergence of Some Statistics with a Mixing Sample
(13.1.17)-(13.1.20) and (13.1.25) together imply the conclusion of Theorem
13.1.3.
13.2 Error variance estimations in linear models
Consider a linear regression model
Уг = х'ф + е» г = 1,2,·..,
where {xi} is a known p-demensional design sequence, {Y{} is a sequence
of observed response, β is an unknown p-dimensional vector, and {e;} is a
random error sequence which is strictly stationary, with
Eei = 0, σ2 := Ee\ > 0, i/ := Ee\ < oo. (13.2.1)
On the basis of the residual sum of squares, estimation of σ2 is
-2(»)=^γ{ς^-ς(Σ"·!;4-)2},
where rn is the rank of matrix Xn = (ж1? · · · , #n)' and stable to rn < ρ
when η is large enough, and (a^l)) is an n-th real orthogonal matrix decided
by design matrix Xn. Put
oo
T = и - σ4 + 2 £ #(e2 - σ2)(β2 - σ2).
i=2
If 0 < г < oo, define random functions Zn(·) of C[0,1] as follows:
Zn(0) = 0, Zn(i/n) = (i - π)(σ2 - а2)/у/пт, г = 1, · · ·, η,
Ζη(·) is linear in , — .
Ι- η n-l
Moreover define the random function Z(·) of C[0, oo] by
Z(t) = ([t)-r[t])(a*([t))-a*).
Lin (1984) and Lu (1986) studied weak invariance principle and strong
approximation for Zn(-) and Z(·) respectively. We shall concentrate our
attention on the sequence {e;} which is <£-mixing. Similar methods can be
used to study other kinds of mixing error sequences.

13.2 Error variance estimations in linear models
361
13.2.1 Weak in variance principle
For any q > 0, define
a(q) = infia : sup sup Eef IA < a]. (13.2.2)
L A:P(A)<v/q4 г J
It is clear that a(q) [ 0 as q | oo. Denote the inverse function of a = a(q)
by q — g(a). Obviously, it is non-increasing. Let a — d(£) be the solution
of the equation a/q(a)b = £. Putting #(£)) = d(21/4/b), we have #(£)) j 0
as b —> oo. Let fn(| oo) be maximum integers satisfying t^g^n n1'4) =
o(l). It is easy to see the existence of such tn. Lin (1984) showed:
Theorem 13.2.1. Let the random error sequence {ei} with (13.2.1)
be strictly stationary and φ-mixing. Suppose that mixing coefficients φ{η)
satisfy
(i) Σ~=ιΨ1/2(η)<οο.
Then τ < oo. If, in addition, τ > 0 and
(ii) nt-^tn) =o(l),
then
Zn => W as η —> oo.
In order to prove the WIP, we need some lemmas.
Lemma 13.2.1. Let {oqn\i — 1,···,η} be a sequence of series of
random variables, which are independent within each series, satisfying
Ea<fl)4I(\a<T)\>q)<a(q), i = l,-,n, (13.2.3)
where a(q) is defined in (13.2.2). Then we have
b*P(\X\>b) = 0(g(b)) as b -> oo (13.2.4)
uniformly in the class Τ of the random variables with the form of X —
Σα=ι 'ai^i where ak satisfy Σ%=1 a\ < 1.
Proof. Let fi :— fin and F{ :— Fin be the characteristic function and
the distribution function of α\η* respectively, f\ ' k-ih derivative of fi. We
first show that
ш) = Т,-Чг^гк+о^ ™ί^° (13·2·5)
k=o k'

362
Chapter 13 Convergence of Some Statistics with a Mixing Sample
uniformly in г = 1, · · ·, п. То this end, write
/«О - Σ ^TT-t4 = "if / (1 - «'"V^x), (13.2.6)
fc=o κ· 4· •/-0°
where |0| < 1. Put t = ±a(q)/q5. For \x\ < q,
\l-eMx\<\tex\<a(q)/q4.
Hence (13.2.6) and (13.2.3) imply that
am- Σ ^l^iLj'-^'^'W
fc=0
+ / 2ж4^(ж)} <t4a(q)/8
J\x\>Q }
l\x\>q
for г = 1, · · ·, η. Note that a(g) is independent of г and a{q) [ 0 as g —► oo.
Uniformity of (13.2.5) is proved. (13.2.5) can be rewritten as
4
log fi(t) = Σ Μ' + »(*)> i = 1, · · ·, n, (13.2.7)
where gi(t)/t4 = 0(a(g)) as £ —► 0(g —► oo) uniformly in г = 1, · · ·, n. Let
/ and F be the characteristic function and the distribution function of
X = Σ?=ι aia\n) respectively. Then (13.2.7) implies
η An η
logf{t) = Σ,Ιοεfi(ait) = Σ{Σ,**ία()#+Σ,9*Μ
г=1 ji=l г=1 г=1
where
η η
t=l t=l
And further
/(*) - Σ ^!P-tk + °(*5 + d(*)*4)
fc=0

13.2 Error variance estimations in linear models
363
since t/d{t) —► 0 as t —► 0 by the definition of d(-). Then, uniformly in /
0(d(t)) = /<4)(0) - {/(4i) - 4/(2t) + 6/(0)
\4
4/(-2ί) + /(-4ί)}/(2ί)4
0itx „—itx.
/г /егъх — е~гъх\ 4
x'dF(x) - J ( ^—) dF(x)
■/('-(r)>w
> i / х4^(ж) > >P(|*| > ъ)
2 У χ >ь 2
/|x|
provided (M)4 > 2, which implies (13.2.4).
Remark 13.2.1. Similarly, if we assume that for some integer m > 0
and <5, 0 < δ < 1,
max #|ahm+*'(kW| > <?) — 0 as g - oo
1<г<п
instead of condition (13.2.3), we have
P{\X\ > b} = o(b~m-^) as 6 -+ ex)
uniformly in Τ defined in Lemma 13.2.1.
Lemma 13.2.2. Under the conditions (i) and (ii) in Theorem 13.2.1,
for any ε > 0
nP(\X\ > η1/4ε) = o(l)
uniformly in rand&tn variables with form X — Σ£=1 а&е& where Σ%=ι α\ <
1.
Proof. Put
(2j+l)tn
^n = [ni~V2], 0 = Σ cLkek,
k=2jtn+l
2(i+l)*n
% = Σ afcefc' j = 0,1,···,/ιη - 1,
fc=(2j+l)in+l
η
fc=2/intn+l

364
Chapter 13 Convergence of Some Statistics with a Mixing Sample
For any q > 0, we are going to estimate
(2i+l)t„
в = {ко|>5^у2(Е«!)%}.
fc=2jt„+l
For brevity, we only consider the case when j = 0, and put the event
k=l
We have
Ефв = Σ a\Ee\lB + £ a\a\Ee\e\lB
k=l p^q
арадЕерея1в + /^ арадагЕереяег1в
рфя рфяфг
+ У, арадага8Еередеге81в· (13.2.8)
рфяфгфз
Obviously, none of the first two sums in the right hand side of (13.2.8) is
exceeded by
fc=l ~~ ~~
Put Μ — maiXi<k<tn Ее\1в- For the third sum, there is
|Σ 4aqEeleqIB\ < Μ £ |α}| Σ |α,| < Μ#» (£ α|)2.
Similarly the absolute value of the fourth sum has the upper bound
Mtn(Yfj^=1 ol\)2. For the fifth sum, its absolute value has the upper bound
ΜίΙ(Σ)?=ι al)2· Therefore we obtain
Ефв<Ш1*(^Га1)2. (13.2.9)
k=l
Referring to the estimation of E^qIb, it is easy to see that Εξ$ has upper
bound bvt\ (Σΐ=ι αϊ) > which implies P(B) < v/qA. From (13.2.2), we
have Μ < a{q). Inserting it into (13.2.9) yields
-Л2{Т,4)~2Ефв < a{q). (13.2.10)

13.2 Error variance estimations in linear models
365
For £j, j — 1, · · ·, hn, we have almost the same conclusions (for £^η, there
may be a difference of a constant). Let {£j,j = 0,1, · · ·, hn} be
independent random variables such that £'· obeys the same distribution as £j. By
(13.2.10),
(2j + l)tn
{5-i/V/2( ^ αϊ)' '& j = 0,1,··-,Η»}
k=2jtn + l
satisfies the conditions given in Lemma 13.2.1. Choosing (^Liytu+i ak)
as dk and tu η1/4ε as b in Lemma 13.2.1 (ε > 0 is given arbitrarily), we
obtain
^2пР{*« 1/2|Σ^| * *η1/2ηχ/4ε} = О(^пЧ)).
Σ
"i=o
Because of the choice of tn, we have
ηΡ{\Σ%\>η1/Αή = 0^ (13.2.11)
i=o
Furthermore by Lemma 1.2.9
\E exp^n"1/4 ]P 3) - Вexp^in"1/4 ]P £,·) |
j=0 j=0
< (hn + l)^(in) < -nt~V(^n).
By condition (ii) in Theorem 13.2.1, ™-1/4Х^=0£у has the same limit
distribution as n"1/4^^^·. Thus from (13.2.11)
nP{\t,tj\>n1/4e}=o(l).
For 77J, we have the same relation. Combining these two results implies
the conclusion of the lemma.
Proof of Theorem 13.2.1.

366
Chapter 13 Convergence of Some Statistics with a Mixing Sample
Obviously it is enough to prove the WIP. Define Un{·) and Vn{·) by
fn(0) = 0,Vn(0) = 0,
We have
By Theorem 5.1.1
Vn{i/n) = Σ (Σ afkek) /Vnr, t = 1, · · ·, n,
j=\ k=X
[i> — 1 in
, — .
η η-Ι
t/n =>> W as η —> οο.
Hence in order to prove the theorem, it suffices to show that for any e > 0
P{ sup |УП(«)| > ε} -> 0, as га -> oo. (13.2.12)
Since π < ρ, (13.2.12) is equivalent to
г
p{ max IУ" aj^eJ > (ητΫΐΑε1ΐ2\ -> 0 as η -> oo
for any {<4 } satisfying Σ£=ι aL ^ 1· By Lemma 13.2.2, we have
p{ шах|У4\|> (nr)1/^1/2}
fc=l
г
< η max p(| Va!?eJ > (rarWV/2) -> 0 as η -> oo.
~~ 1<г<п Uf-' * I ~~ v 7 J
fc=l
The proof of the theorem is complete.
Remark 13.2.2. When {e^} is a strictly stationary m-dependent
sequence, one can take tn to be a constant, and hence condition (ii) is
satisfied. When {ei} is a bounded sequence, we can take a(q) ξ 0 and £n,
for example, to be [ra2/3]. Hence condition (ii) is also satisfied.
Remark 13.2.3. Lin (1984) also gave the WIP when {e;} is
admixing.
13.2.2 Strong approximation

13.2 Error variance estimations in linear models
367
Using the strong appximation result for a (^-mixing sequence (cf.
Theorem 9.1.1), Lu (1986) showed the following theorem.
Theorem 13.2.2. Let the random error sequence {e^} with (13.2.1)
be strictly stationary and φ-mixing. Suppose that E\e\\^8 < oo for some
0 < δ < 1 and
φ(η) = o(ni-4(2+e)(i0-*)/**) as η-^oo (13.2.13)
where 0 < θ < 1 and ε > 0. Then
Z(t) - W(t) = 0(i1/4(log *)9/4+e) а.з.
In order to prove the theorem, we need a lemma:
Lemma 13.2.3. Under the conditions of Theorem 13.2.2,
X = o{n1'*) а.з.
where X — Σ£=1 а&е& with Σ2=ι αΙ < 1·
Proof. Put
dn = [nM/4(10-*)], /*n = [n/2dn],
(2i+l)dn 2(i+l)d„
fi = Σ afcefc' ^i = Σ afcefc' j = 0,1, · · ·, /in - 1,
fc=2jdn + l fc=(2j+l)dn + l
η
fc=2undn + l
Similarly to (13.2.9),
si&iw<s(i!EiW)
k=l
< Σ |α,|8+^|β,|8+ί + £ (Σ Ы>/-*+^|е*Г|е/-^|)
fc fc/ji г=1
+ ···+ Σ lafci---afc8lKh^|efci---efc8||efc9|6|
fc=l

368
Chapter 13 Convergence of Some Statistics with a Mixing Sample
Let {&Λ = 0,1, · · · ,/in} be such independent random variables that £'·
obeys the same distribution as f^. Put
10-6 ,(2j+l)dn .
tll , 2(8+6) / V"^ 2\ I £l
k=2jdn + l
and
ai = { Σ 4) ■
k=2jdn+l
Then
hn η
Σαγ<Σα\<1.
3=0 k=l
Applying Remark 13.2.1 toI = EjZo"j£" and b = £dn(10~*)/2(8+*V/8,
we obtain
^{|Σφ-1/8}
3=0
10-< hn 10-6
= P{^2(8+£)|£^| >ed„2(8+4)ni}
= o^-^n"1-6/8) = oin"1-^1-*)/8). (13.2.14)
By у?-mixing property,
\Ρ{ξι+ξ2<χ}-Ρ{ξ[+&<χ}\
< J \Ρ{ξι <x- u\& = u} - Ρ{ξ1 <χ- u}\dP{& < x}
< f(dn),
and hence
\ρ{Σ,*><*)-ρ{Σ.$ϊ*}\
j=0 j=0
< 0(hn<p(dn)) = 0{η~ι-ε). (13.2.15)
Combining (13.2.14) with (13.2.15) implies
Ρ{\Σϊι\>εη1/8}=0(η-ι-η.
3=1

13.3 Density estimations
369
By the Borel-Cantelli lemma
|Efc|= ofn1/8) a.s.
j=o
Similarly
/in-l
The lemma is proved.
Proof of Theorem 13.2.2.
Write
M r[t] / W /r η χ 2
fc=l i=l j=l
r[t) [t] 9
Prom Lemma 13.2.3, we have
Μ 2
(Σα«Ι)βί) ^°(^/4) a's- as*-^oo. (13.2.16)
i=i
By Theorem 9.1.1, there exists a Wiener process {W(£),£ > 0} such that
for any ε > 0
X(i)-W(i) = 0(i1/4(logi)g/4+e) a.s. ast-^oo. (13.2.17)
Combining (13.2.16) with (13.2.17) implies the conclusion of the theorem.
13.3. Density estimations
Let {Χη,η > 1} be a sequence of i2d-valued random variables with a
common density function f{x). In general, there are two kinds of
estimations for f(x). The first is the so-called kernel estimation, which is defined
by
x — X;
Ш = μί)-^4^), (13.3.1)
i=i hn

370
Chapter 13 Convergence of Some Statistics with a Mixing Sample
where the window width hn [ 0 as η —> oo. Another kind is the so-called
nearest neighbor estimation, which is defined by
fn(x) = kn/{n\S(x,an(x))\}, (13.3.2)
where /cn, 1 < kn < n, is the given integer, an(x) is the distance from χ to
the kn-th closest X{ (in Χχ, · · · ,Χη), S(x,a) is the hypersphere of center
χ having the radius a and |S(#,a)| = L(S(x,a)), where L denotes the
Lebesgue measure in Rd.
In this section we always assume {Xn} is (^-mixing. Other kinds of
mixing sequences can be studied similarly.
13.3.1 Kernel estimation
Many authors, such as Lin(1983), Masry and Gyorfi (1987), Shao
(1990), Cai (1991), Peligrad (1992), Fan and Xue (1993) have studied
limit behavior of the kernel estimation (KE) of the density function for a
mixing sequence.
Let {ХгцП > 1} be a unvalued (^-mixing sequence with a common
unknown density function f(x) = /(#i, · · ·, ж<г)· Consider the KN fn(x)
defined by (13.3.1). Peligrad (1992) showed the following result:
Theorem 13.3.1. Suppose that D is a compact subset of Rd and f
is continuous on an ε-neighborhood of D. Suppose that К satisfies the
following conditions:
(1) K(-) is a density on Rd,
(2) K{x) <Kx<oo for any χ G Rd,
(8) \\x\\d+1K{x) -> 0 as χ -> oo,
(4) f\\x\\K(x)dx = K2<oo,
(5) K(·) is Lipschitz of order 7 on Rd.
Then
sup \fn{x) - f(x)\ - 0(hn + d}/2 \ogn/{nhdn)l/2) a.s. (13.3.3)
xeD
where dn — expi2j]|^nJ (^1/2(2Z)J. //, in addition, the condition
00
Σ Ψ1/2(2η) < oo (13.3.4)
n=X
is satisfied, then for hn = О ((log2 n/n)1^d+2A
sup \fn(x) - f(x)\ = 0((log2 η/η)χ/(<*+2)) a.s. (13.3.5)
xeD

13.3 Density estimations
371
Remark 13.3.1. If condition 3) is weakened into
(3)' llar^A» -> 0 as χ -> oo,
condition (4) is dropped, and condition (13.3.4) is replaced by
and
we have
lim φ(η) < 1/2
η—>oo
nhn/(dn log2 n) —> oo asn-> oo,
sup |/n(#) — /(ж)| —> 0 a.s. as η —> oo.
Remark 13.3.2. Under independence assumption, the rate of the
maximal deviation of the estimation from the true density is due to Kuelbs
(1976) and its size is O^loglogn)1/2/™1^2^*1))). (13.3.5) improves the
speed of this convergence to (9((log2 n/n)1/^"1"2)).
Remark 13.3.3. Shao (1990) further improved the speed to 0((log n
/n)1/^"1"2)), but a stronger mixing condition, i.e. φ(η) — 0(n"~(2+d)), is
required.
Proof of Theorem 13.3.1.
Obviously by Lemma 2.2.2, under condition (13.3.4), (13.3.3) with
hn = οαΐο^η/η)1^**2)) implies (13.3.5). We prove (13.3.3). Write
sup|/n(aO ~f(x)\
xeD
< sup \fn(x) - Efn(x)\ + sup \Efn(x) - f(x)\
xeD xeD
=:Δχ + Δ2. (13.3.6)
Estimate Δχ first. Because D is compact, we can choose a covering of
D with l(n) balls βχ, · · ·, Β/(η) of centers ίχ, ·· ·, t/(n) having the radius
^-^(n-^^log2^1^;
the number l(n) can be chosen less than 0(h-d(n/(hd log2 n))d^2^).
Without loss of generality, we can assume hn > n~2/d. Hence
l(n) = 0{п2+ы'^). (13.3.7)
Let ж be in D and define
ад-^Е(^)-ЦНг))-

372
Chapter 13 Convergence of Some Statistics with a Mixing Sample
So
Δχ = supinhi^Snix).
xeD
By condition 5), for every χ € Bk, we have
\Sn(x) - Sn(tk)\ < CnRlh-dl2~i < C(nlog2 η)1'2 (13.3.8)
for some С > 0. And by Lemma 2.2.2, there «xists G > 0 such that
ESl(x) < Gndn. (13.3.9)
Moreover by condition 2)
<2K1(Gndnhi)-1/2=:Cn.
Now let A(> C) be a positive number specified later on. By (13.3.8)
we have
P\ sup \Sn(x)\ > 2A(nlog2n)x/2)
l(n)
< £(p{ sup \Sn(x) - Sn(tk)\ > ^(nlog2^1^}
+ P{\Sn(tk)\>A(n\og2n)1/2})
<l(n) max P{\Sn(tk)\ > A{n\og2 n)1/2}. (13.3.10)
l<k<l(n)
Estimate the probability of the right hand side of (13.3.10). It is clear that
for any η > 0, 0 < η < 1/2, there are ρ > 1 and A > 0 such that for n > ρ
φ{ρ) + max P{\Sn{tk) - Si(*fc)|/((?ndn)1/2 > A} < η.
1<г<п
By a combination of (2.2.18) and (2.2.19) in Lemma 2.2.7, we have
P\ max iSjit^l/iGndn)1/2 > χ + 2A + 2pCn\
{ l<j<n >
< ~^—P{ max \Sj(tk)\/(Gndn)1/2 > x\. (13.3.11)
1 — 77 ^i<j<n J
Let Bn — 2A + 2pCn and Mn = maxi<j<n |5j(tfc)|/(Gi^n)1/2. Obviously,
for any an > 0
£exp(anMn) < exp(anBn) + an / ехр(апж)Р(Мп > ж) cfo.

13.3 Density estimations
373
After changing χ to χ + Bn, by (13.3.11) we get
Eexp(anMn) < exp(anBn) + ——-Eexp{an(Mn + Bn)}.
Letting an — (2Bn)~1\og^~1 — 1) yields
Eexp(anMn) < ((τ?"1 - l)"1/2 - (η'1 - l)"1)"1 =: 9{η).
Put α = infna„ = (4A)~l log^~l - 1). Then
Piailognr^GndJ-WlSMl > 3d/(27) +4}
< 9{η) exp{-(3d/(27) + 4) logn}
= 0(η-(3^>+4>) (13.3.12)
uniformly in к as n —> oo. Putting A = 2((3d/27 + 4)G1/2/log(iT1 -
l))1/2, which implies A = {M/2^ + A)Gxl2/a. From (13.3.7), (13.3.10) and
(13.3.12) we obtain
oo
V PJsup \Sn(x)\ > 2A(ndn\og2 n)1/2\ < oo. (13.3.13)
n=i L^^ J
Therefore
//dnlog2n\i/2x
Al = °U nfed j J a's- asn-^oo.
As for Δ2, by the well-known Bochner-Parzen theorem (cf. Parzen
1962), under the conditions of the theorem, we have
Δ2 — 0{hn) as n —> 00.
The proof of the theorem is completed.
13.3.2 Nearest neighbor estimation
Chai (1984) studied strong consistency of nearest neighbor estimation
(NNE) of the density function for a mixing sequence. Using the improved
Bernstein inequality, Lemma 12.4.1, Chai's theorems hold under the weaker
mixing condition.
Theorem 13.3.2. Suppose that condition (13.3.4) ™ satisfied and kn
in (13.3.2) satisfy
kn —> 00, kn/n —► 0 as n —> 00. (13.3.14)

374
Chapter 13 Convergence of Some Statistics with a Mixing Sample
Then kn/ y/n —► oo implies
fn(x) -A f{x) a.s. χ G Rd(L); (13.3.15)
and Σ^ι exP(~c^n/n) < °° for апУ c > 0 implies
fn(x) -> f(x) a.s., a.s. χ G Rd(L). (13.3.16)
Proof. Denote V& the volume of an unit ball in Rd, μ and μη the
distribution of X\ and the empirical distribution of Χχ, · · ·, Xn respectively.
For any given ε > 0 put
bn(x) = (f(x) + £)Vdn/kn,
b'n{x) = (f(x) - e)Vdn/kn,
Sn(x,b) = S(x,b-1/d(x)),
Sn(x,b') = S(x,b'-1/d(x)).
Then
P{\fn(x)~f(x)\>e}
< P{fn(x) - f(x) > ε} + P{fn{x) - f(x) < -ε}
< P{/in(5n(x,b)) -μ(5η(χ,6)) > —
η
-М5п(Ж,Ь))} + РК(5„(х,Ь'))
-μ(5η(*,6')) > — -/*(£„(*, δ'))} (13.3.17)
η
(If /(ж) < ε, the second term of the right hand side of the first and the
second inequality sign disappears). By the well-known Lebesgue density
theorem we have that as η —► oo
μ(5η(χ,6))/|5η(χ,6)| - f(x) a.s. χ G i?d(L),
/i(5n(x,b,))/l^n(^,i>,)| - /(ж) a"*, x G i?d(L).
Let the exceptional sets be denoted by D and D' respectively. Put Ε =
Dc Π D/c. Then for any χ £ Ε and η large enough,
M5„(x,6))<^(/(x) + |)/(/(x) + e),
μ(5η(χ,?>'))>^(/(^)-|)/(/Η-ε).

13.3 Density estimations
375
Put a(x) = e/(2(/(ar) + ε)) and a'(x) = e/(2(/(ar) - ε)). For any χ € Ε
P{\fn(x)-f(x)\>e}
< P{^n(Sn(x,b)) -M(S„(ar,b))| > —a(x)}
η
+ P{^n(Sn(x,b')) -/*(5„(ж,6'))| > —α'(«)}
η
=: /ni + In2.
Let & — I{X% £ £п(#5 £))) — /z(Sn(#, £))), г = 1, · · ·, η. Then using Lemma
12.4.1 we have
r k2 ϊ
Jni <2exp\-c^a2(x)\,
and similarly
ι 2
/n2<2exp{-c^a,2(x)|.
They imply (13.3.15) if kn/y/n -> oo and (13.3.16) if Σ™=ι exp(-c*£/ra) <
oo for any с > 0.
Remark 13.3.4. By a finer analysis, we can obtain a rate of a.s.
consistency, i.e.
fn{x) ~ f{x) = o(r~x) a.s.
where rn = η^,Ο < β < l/(2(d + 1)), if /(·) satisfies the local Lipschitz
condition on χ and /(ж) > О and conditions (13.3.4) and (13.3.14) are
satisfied.
Next we consider uniform strong consistency in the case of d = 1. At
this time, fn{x) — кп/(2пап(х)), х G R. We need some lemmas. Let
F and Fn be the distribution function of X\ and the empirical
distribution function of X\, · · ·, Xn respectively. Define the empirical process of
{Xn,n>l}:
R(s,t) = [t](F[t](s) - F(s)), seR,t>0.
Lemma 13.3.1. (Berkes and Philipp 1977) Suppose that
ψ{η) = 0(n~b-6) for some δ G (0,1/4). (13.3.18)
Then there exists a version K(s, t) of a Kiefer process such that
supsup|#(s,*) - K(F(s),t)\ = 0(T1/2(logT)-A) a.s.
t<T seR
for some λ > 0.

376
Chapter 13 Convergence of Some Statistics with a Mixing Sample
Lemma 13.3.2.(Csorgo and Revesz 1981) For a Kiefer process K(s, t),
limsup sup \K(s, t)\/(t log log t)1/2 = 1/^2 a.s.
t^oo 0<s<l
Theorem 13.3.3. Suppose that (13.3.18) is satisfied and {kn} satisfies
kn/n -► 0 and kn/(n\og\ogn)1/2 -► oo. (13.3.19)
And suppose that f is uniformly continuous in R. Then
sup \fn(x) ~ f(x)\ -► 0 a.s.
X
proof. By Lemma 13.3.1, for any ε > 0 there is a constant a > 0
such that
P(An,i.o.) <e, (13.3.20)
where An — {sup^. \R(x,n) -K(F(x),n)\ > an1/2 (log n)~x\. Similarly, by
Lemma 13.3.2 for any ε > 0 there is a constant b > 0 such that
P(Bn,i.o.) <e, (13.3.21)
where £?n — jsupo^^ |if(s,n)| > b(nloglogn)1/2}. Put
2п(/(*)+е)' ην 7 2n(/(x)-e)
and
Sn(#, dn) = (χ - dn(x),x + dn(x)),
Sn(x, d'n) = (x- d'n{x),x + d'n(x)).
Then similarly to (13.3.17)
P{\j{sup\fn(x)-f(x)\>£}}
<P{U \J{Vn(Sn(x,dn))
n>m x
-/i(5n(x,dn)) > — - μ(3η(χ,άη))
+ ^{U U K(5n(x,0)
n>mcc:/(cc)>e
-/*(sn(*X))>-^-/*($„(*, <))}}
=·' Jml + «Λη2· (13.3.22)

13.3 Density estimations
377
By uniform continuity of /, Μ :— supx f(x) < oo, and further for any χ
and large n,
μ(5»(*, «U)<^(/(*) + f )/(/(*)+ ^
η 2
η -M^n^,dn^> n ·2(/(χ)+ε) ^ n ·2(Μ + ε)
Μ^(*.<)) >-(/(*)-f )/(/(*)-*).
μ(5η(χ,<)) - ^ > *"
= :Ρη>
η - η 2(/(ζ)-ε)
>
κη ε
~η ' 2(Μ - ε)
=: <7η, as /(ж) > ε.
Therefore
«Ληΐ < ^{ |J |sup|/in(5n(x,dn)) -/i(5n(x,dn))| >pn}}
<2Р{д{^р|^»И-^)1>у}}
< 2Ρ
п>т
{υ{
sup
ι R(x, n) — K(F(x),n) ι pi
>
η
pf}}
+ 2P{U{ Ж№^}}
n>m
— 2 J(1) + 2 J(2)
By condition (13.3.19) we have
^/npn(logn) —► oo as η —► oo.
Hence it follows from (13.3.20) that
rW
|Д(ж,п) -K(F(x),n)i n^pnilogn)*
n1/2(logn) λ
< P{ (J An\ —> 0 as m —> oo.
Similarly (13.3.21) implies
i2J < P{ (J Bn} -+ 0 as m -^ oo.
n>m

378
Chapter 13 Convergence of Some Statistics with a Mixing Sample
Thus Jrni ->0asm-^oo. Moreover
Jm2 <P{\J {sup|/in(5n(x,<)) -μ(5η(χ,<))| > qn}}.
n>m
In the same way as for Jmi, we have Jm2 ->0asm-^oo. Then it follows
from (13.3.22) that for any ε > 0
P{ U {SUP \f"(x) " /(ж)1 >ε}} ~^° ^ m "^ °°·
n>m
This completes the proof of the theorem.
Remark 13.3.5. If / possesses a bounded second derivative then for
kn = [n7/10] and any Cn —> oo we have
sup|/n(x) - /(a:)| = ofr-^Ooglogn)x/2Cn) a.s.
Remark 13.3.6. In Yu (1993) a simple and useful nonparametric
estimator of a density /(ж) based on a sample Χχ, · · ·, Xn has been defined.
If m = mn is a positive integer, the nonparametric estimator fn(x) of f(x)
is defined by dividing 2ran by η times the length of the smallest interval
containing χ which consists of 2ran of the η observations in which the
half of them lie on the left side of χ and the other half on the right side.
Formally,
—— - -, X G pf(mn+j)>^(mn+.7+l))5
for j = 0,1,···,,η-2τηη;
0 X < X(mn) ОГ X > X(n_mn+1),
where (X(X), · · · ,X(n)) is the order statistic of (Χχ, · · · ,Xn). Yu (1993)
showed that in the case of i.i.d. observations /n(#) converges to f(x) (in
probability and a.s.) under some conditions. Yu (1995) studied that the
rate of strong uniform convergence for the estimator defined as above when
the observations satisfy a (^-mixing or an α-mixing condition.
fn{x) = <

Chapter 14 Strong Approximations
for Other Kinds of
Dependent Random Variables
The almost sure invariance principle for sums of weakly dependent
random variables has been given by Philipp and Stout (1975), such as
lacunary trigonometric series, (^-mixing or α-mixing sequence of random
variables, Gaussian sequence and additive functional of Markov chains .
The strong approximations of sums of (^-mixing and α-mixing sequences
have been studied in Chapter 9. In this chapter we shall investigate the
strong approximations of sums of other kinds of dependent random
variables, including lacunary trigonometric series with weights, a class of
Gaussian sequences and additive functional of Markov processes. All of these
improve essentially and comprehensively the results in Philipp and Stout's
monograph (1975).
14.1 Lacunary trigonometric series with weights
Let {η^, к > 1} be a sequence of positive real numbers with
nk^/nk>l + q/kr (14.1.1)
for all к and some q > 0, 0 < r < 1/2. It is said to be a lacunary sequence
when r = 0. Let {а&, к > 1} be a sequence of non-zero real numbers. Put
x=lib°i fi=iib*i (14л·2)
Suppose that An —> oo and that there exist constants <5,/3 with 0 < δ <

380
Chapter 14 Strong Approximations for Other Kinds
l,/3 > 0 such that
ak = 0{A\-6), (14.1.3)
A2k = 0(Ak), (14.1.4)
hP = 0(Ak). (14.1.5)
In this section, ([0,1),β,Ρ) denotes a probability space, where В
consists of the Lebesgue measurable sets of [0,1), and Ρ is the Lebesgue
measure on B. We consider the trigonometric series
η
S{A2n,u) = Y^akcos2nnku, ω G [0,1). (14.1.6)
For t > 0, put
S(t,u) = S(Alu), ifA2n<t<A2n+1, (14.1.7)
where Ao = 0.
The strong approximations of S(t) by a Wiener process W(t) were first
discussed by Gaposhkin (1966). Later on, in the case of r — 0, Philipp
and Stout (1975) proved the almost sure invariance principle for lacunary
trigonometric series with weights under condition (14.1.3) and obtained
the approximation order of 1/2 — λ for each λ < δ/32. In a special case of
unweighted summands, they obtained the approximation order of 5/12 +λ
for each λ > 0 and conjectured that the constant 5/12 would be replaced by
1/3. Sun (1984) showed this fact. Shao (1987) improved all these results
comprehensively. He studied the general case of lacunary trigonometric
series with weights and pointed out that Sun's order is not the best possible,
and obtained, when the case of r = 0, the order of 1/4 which is the best
if one were only to use the Skorohod embedding scheme. Furthermore, in
some particular cases, the logarithmic order is obtained
Theorem 14.1.1. Suppose that conditions (ЦЛ.З)-(ЦЛ.Б) are
satisfied and that δβ > r. Then without changing the distribution of{S(t),t >
0}, we can redefine the process {S(t),t > 0} on a richer probability space
together with a Wiener process {W(t),t > 0} such that
S(t) - W{t) = o(f2-^i+rl^ log21) a.s. (14.1.8)

14.1 Lacunary trigonometric series with weights
381
Theorem 14.1.2. Under the conditions of Theorem Ц.1.1 and the
assumption of \ak\ non-increasing, we have
S{Al) - W{Al) = o(AZ/W{BW V A'JW)) log2 An) a.s. (14.1.9)
We immediately obtain the approximation order for lacunary
trigonometric series with unweighted summands from the above general theorems.
Corollary 14.1.1. If a& = 1, then we have
S(t) - W(t) = o(*(1+2r)/4log2*) a.s.
In the case of r — 0 the result (14.1.10) is an essential improvement
of Theorem 3.1 in Philipp and Stout (1975), the order of 1/4 is the best
possible provided S(t) is constructed via the Skorohod embedding method.
Corollary 14.1.2. Suppose that the condition of Theorem 14-1.2 and
condition (ЦЛ.5) are satisfied (for some β > 0), and Βχ = O(l), \ak\ 1 0.
Then when r = 0 we have
S(t)-W(t) = <3(log2*) a.s.
The proofs of Theorems need the following lemmas.
Lemma 14.1.1. Let £1, · · · ,£n be a sequence of random variables. Put
к
5* = Σ&·> Μ*= max |Sj|.
Suppose that there exists a sequence {c&} such that for all 0 < г < j < η
Е^-3{\2< Σ сь
i<k<j
then
г=1 &
for each к < n.
The proof refers to Theorem 2.4.1 in Stout (1974).

382
Chapter 14 Strong Approximations for Other Kinds
Lemma 14.1.2. For any ν > 0,
§f=<H· <ΐ4ΐ·ιο>
j>k J k k
Proof. (14.1.4) implies that there exists a constant С > 0 such that
Α+, = θ((ψ)4·)
for all /с, j > 1. And by η^+ι/η*. > 1 + q/kr', we have
where we have used the well-known formula
η i—r
Yk~r = - + u + 0(n-r), 0<r<l/2,
*-** 1 — г
fc=l
where и — ϊχ(γ) is a constant. Hence
f^nt УпккЛ^к-> П 2(1 -τ))> PV2(l-r)^
ynk >
Thus (14.1.10) is proved. The proof of (14.1.11) is similar.
Lemma 14.1.3. Let W(t) be a Wiener process and {tn} be a sequence
of random variables. Suppose that there exists a sequence {&„} of real
numbers with bn = o(n), such that
tn-n = 0(bn) o.s., (14.1.13)
then
W(tn) - W(n) = 0{blJ2 log1/2 n) a.s. (14.1.14)

14.1 Lacunary trigonometric series with weights 383
Proof. (14.1.13) implies that there exists a constant С such that
\tn ~ n\ < Cbn a.s.,
hence
\W(tn)-W(n)\< sup sup \W(t + s)-W(t)\ a.s.
0<t<n-Cbn 0<s<2Cbn
In terms of a well-known result (cf. Theorem 3.2B in Hanson and Russo
1983), we have
sup sup \W(t + s)-W(t)\
0<t<n-Cbn 0<t<2Cbn
= o((bn(log^^ +loglog(n + C6n)))1/2) a.s.
(14.1.14) follows from the above relations.
We now define an increasing sequence Tk of σ-fields as follows. Put
ρ = 2/β + 4. For each integer /с, let r& be the largest integer г such that
21' < A\nk. (14.1.15)
We define Tk to be the σ-field generated by the intervals of the form
Uvk = [v2~rk, (v + l)2~rfc), 0 < ν < 2rK
Putting £fc(u;) = аксоз2кпьи, Хк(и) = E^^F^.), we obtain by (14.1.15),
for each fc, j > 1
E(tk+j\Fj) = 0(ak+j(l A Apjnj/nk+j)). (14.1.16)
Lemma 14.1.4. We have
oo
ΣΙ&-^*Ι = 0(1). (14.1.17)
k=l
Proof. We have from (14.1.15)
\tk-Xk\ = 0(Ak-pak) = 0(Ak-*+1).
Hence (14.1.17) follows from (14.1.5).

384
Chapter 14 Strong Approximations for Other Kinds
Lemma 14.1.5. We have
£ EX]-A2N = 0{\). (14.1.18)
1<3<N
Proof. Noting that
Εξ] - Oj/2 = a2j sm2nnj/2nnj,
j<N j<N
and
£ [EX] - Εξ]) = 0( £ a]Ajp) = O(l),
j<N j<N
we get (14.1.18).
We can now represent Xj as
Xj = Yj + uj - uj+u (14.1.19)
where {Yj, Tj\ is a martingale difference sequence and u\ = 0
oo
Uj = Σ Е(Х^к\Ъ-г), j > 2. (14.1.20)
fc=o
Lemma 14.1.6. We have for all j, 1 < j < η — 1,
£ \Е(Х{Хк\^)\ = Сп2гА2п~261оЕ2Ап, (14.1.21)
j<i<k<n
where constant С does not depend on j and n.
Proof. By the definition of X&, we admit
Е{Х{Хк\^) = E{X£k\Fj).
Write
2ri-l
EfaXifo) = Σ I(Uvj)bv.
v=0

14.1 Lacunary trigonometric series with weights
385
We have
2~rj 2Ti~rJ-l fv2~rJ+(l+l)2~ri
-bv = } 2Ti \ cos27rriktdt
ak^i f^ Jv2~rJ+l2~ri
rv2~rJ +(l+l)2~ri
• / cos2nriitdt
Jv2~rJ+l2~ri
„ .! sin 2ттщ2~п~1 sin 2nnk2~ri~1
— Ψιτ!
{2тт)2щпк
2ri~rJ-l -
• 5^ (cos2n(nk - щ){у2'г* + {l+ -)2~ri)
1=0
+ cos27r(nfc + тц)(у2-г* + (l + i)2~r<)).
Using the equality
Tl—l · /r%
Σ, 14 sin an/2
cos(av + b) = — -.— cos(o + a[n — l)/2),
л sin α ι δ
where a and b are any real numbers with sin(a/2) φ 0, we obtain
2-r*bv _2ri+1 sm2nni2-ri-1 sm2Knk2-ri-1
акщ (2п)2щпк
·(- τ-5 4 rcos2^nfc-пЛ(г> +- 2 rj
Vsin2^nfc-ni)2-r<-1 V yV 2/
Hence
6ν = 0(α*α,·2Γ'(1 + 2~г1щ)/пк)
for all г, A:. And we also have for г, к with η& + щ < 274""1,
bv = 0(аксц(1 Л 2rV(nfc ~ η,·)).
For each г : j < г < η, take fco(i) = тах{/с : (rifc + щ) < 2ri~x} Л (n — 1).
(14.1.12) and (14.1.15) imply that k0(i) - г = 0(ir log Д·), hence
j<z<fc<n
' 2r^ пг·
ο(Σ Σ 1ад|(- + -)
n-1 fco(0 2^·
+ Σ Σ ι^Κ^^ζττ))· (14-L22)
i=j+ik=Zi v nfc~ni'

386
Chapter 14 Strong Approximations for Other Kinds
From Lemma 14.1.2 and (14.1.15), the first part of the right hand side of
(14.1.22) is bounded by
= 0(Al~V).
Take i(j) = тах{г : 2rJ > пгГ3г} Л (η - 1). Then (14.1.12) and (14.1.15)
imply that i(j) — j = 0(jr log Aj). Again from Lemma 14.1.2, the second
part of (14.1.22) is bounded by
i(j) n~X k0(i) r
^-24(Σ Σ ι+ΣΣ-^)
i=j i<k<k0(i) i>i(j) k>i K г
П—1 tyrj
= o(Al-*(n*\o£An+ £ ^-г2"))
Ч -^ ·( ·\ ni+l J '
= оЦ-^(„-1о^д, + ^*1£))
V V ni(j)+l "
= 0(n2rA2n~2S log2 An).
The lemma is proved.
Lemma 14.1.7. We have
Uj = 0(jr A]-6 log Aj). (14.1.23)
Proof. Using (14.1.16), (14.1.20) and Lemma 14.1.2, we find
3 = Y^E(£k+j\fj-l)
oo
k=0
•г л1-6 ]
= 0(fA)-u log Aj).
Lemma 14.1.8. For each к,п(к < η),
£ Е(Хгип+1\Ъ) = 0(А2п-26п2г1оё2Апу, (14.1.24)
k<i<n
Σ E(XiUk\Tk) = 0(A2n-2Snr log An). (14.1.25)
k<i<n

14.1 Lacunary trigonometric series with weights
387
Proof. By (14.1.20) we have
OO
j=0
OO
= Ο^Α]-δη+1(1ΑΑ^ητ/η3+η+1)).
j=0
Hence
η oo
Σ Ε(Χ<ηη+1\Κ)=θ(Σ Σ(1Λ^?ηί/ηί+»+ι)^ΐη+ι^~ί)·
i=k+l i=k+lj=0
Taking ко = [corT log Лп], where со will be specified later, we have
η oo
i+n+i
i=k+lj=0
η ко п—ко оо др
- Σ Σ^/+η+ι+ Σ Σ.,1 * 4/+П+1
i=n-fc0+i y=o i=fc+l i=0 A^+n+X
η oo jp
t=n-fco+li=fco + l '^+η+1
τι—fco jp r
<=fc+i n"+1
+ Σ ^(*o + n+l)M-L+1)
=ο(η>^'-^ρ(<"-^': --"%)+*w
+ ^n fcoexp( ^-^ ,))
= 0(^-5n2rlog2An).
The last equality holds so long as we take cq sufficiently large. This proves
(14.1.24); the proof of (14.1.25) is similar.
Lemma 14.1.9. We have
Σ EYf -Al = 0(n2rAn~26 log2 An). (14.1.26)
3=1

388
Chapter 14 Strong Approximations for Other Kinds
Proof. Since {Yj} is a martingale difference sequence, we have
±EY? = E(±Yj)2
3=1 j=l
= Ε{Σ xif+2Eun+i Σ xj + Eu2n+v
I2
j=l j=l
Hence (14.1.26) follows from Lemma 14.1.5 - Lemma 14.1.8.
Lemma 14.1.10. Under the assumption δ β > r, we have
Σ У? -А2п = 0(A2n-6nr log3 An) a.s. (14.1.27)
j=l
Proof. Note that (14.1.5) and δβ > r imply nr = 0{A8n). Applying
Lemma 14.1.1, the Borel-Cantelli lemma and the subsequence method, we
need only to show that for any 0 < m < η
E( Σ (У/-Е1?))2 = о(( Σ ΕΥ})Αΐ~26η*log2An), (14.1.28)
m<j<n m<j<n
where the constant implied by О does not depend on ra, n. Observe that
Yf-Щ'У'
m<j<n
E( Σ V?-EYf)f
Σ E{Yf-EYff + 2 £ ВД2-Я1?)(Е Υ?),
m<j<n m<k<n k<i<n
where
£ Ε{Υέ-ΕΥΪ)(Σ Υ2)
<k<n к<г<п
= ς вд?-яп2)(Е tf
т<к<п к<г<п
= ς £(п2-я#)(Е х? + 2 Σ κ*
т<к<п к<г<п k<i<j<n
+ 2 ]Г Xi(Un+l - Щ+l) + (Un+i - Uk+1)2)
k<i<n
=: Σ (h(k) + I2(k) + h(k)+h(k)).
m<k<n

14.1 Lacunary trigonometric series with weights 389
By Lemma 14.1.6 - Lemma 14.1.8, it follows that
max 1{(к) = 0(n2rA2n-26EY£ log2 An). (14.1.29)
2<г<4
We now show that (14.1.29) holds for I\(k) as well. Since
| Σ χϊ- Σ φο(ΐ),
k<i<n k<i<n
it suffices to show that (14.1.29) holds for E(Y£ - EY%) (T,k<i<n &) ·
Noting that Yk is .^-measurable and writing
2rfc-l
η = Σ bHPik),
г=0
we have by the definition of Tk
E{Y2-EY2){^ ξ])
j=k+l
2rfc-l η /.(i+l)2"rfc
" ~ * '* мг+ι,μ 'fc
Σ d2 Σ /. α2 cos2 2nrijtdt
i=o j=k+iJi2~rh
2rfc-l η «!
~2_Γ"(Σ rf?) Σ / o2cos227rnjtdt
= Σ <*? Σ α? ^ со84ттДг+-)2 **
г=0 j=fc+l
2гк-\ п a2
= θ(2-'(Σ ^)( Σ ^+ Σ α,2(ΐΛ^ηΛ·)))
г=0 j=fc+l ■* j=fc+l
= 0(A2n-2SnrEY*logAn).
This proves that (14.1.29) holds for h(k). For ЯУД we have
EY£ = 0(EYk2A2~2Sk2r log2 Ak).
This proves (14.1.28), and the lemma follows.

390 Chapter 14 Strong Approximations for Other Kinds
Lemma 14.1.11. We have
η
^(^(F/I^i) - Yf) = 0(nrA2n-6 log3 An) a.s. (14.1.30)
Proof. Put Rj = Yf - E{Yf \Tj-i). Then {Rj,Tj} is a martingale
deiFerence sequence, and we have
ERl = 0{EY£) = 0(EYk2 A2k~2Sk2r log2 Ak).
The proof is verbatim as that of Lemma 14.1.10.
By a martingale version of the Skorohod representation theorem, there
exists a probability space, on which a Wiener process and a sequence of
non-negative random variables T{ are defined such that
{w((j2Ti),™>\} and {^2Yj,m>l}
j<m j<m
have the same distribution. Hence on the new probability space, without
loss of generality we can redefine {Yj} by
Yj = w(J2T^-w(^Ti)
i<j i<j
and can keep the same notation. Write
к
£m = a{w(5^25),fc<m},
3 = 1
m
Λη = ψ(ί),ο<ί<5;Γ,·}.
It is clear that Cm С Αγη,πι > 1, and Tj is Aj- measurable. By the
embedding theorem, for every j > 1, ETj = ΕΥ? , and
Е<?№-х) = E{Y}\Aj-{) = Е<у}\С;-Х) a.s. (14.1.31)
Moreover, for any ν > 1 there is a Cv such that
E\Tj\v = CvE\Yj\2v. (14.1.32)
Lemma 14.1.12. We have under the conditions of Theorem ЦЛЛ
^ Tj -A2n = o(A2n-i+r/P log3 An). (14.1.33)
j<n

14.1 Lacunary trigonometric series with weights
391
Proof. Write
Σ2* - Al
j<n
= Σ& - Epipj-x)) + j;w^-i) - Yf)
+ Σ Yi ~ Al =: h + h + h-
j<n
Put Zj = Tj — E(Tj\Aj-i). Then {(Zj.Aj)} is a martingale difference
sequence and EZJ = O(EY^). By the proof analogous to that of Lemma
14.1.11 we obtain
1г = 0(A2n-6nr log3 An) a.s. (14.1.34)
The lemma follows from (14.1.5), (14.1.27) (14.1.30) and (14.1.34).
Proof of Theorem 14.1.1.
Note that
Σ YJ ~ Σ Хз = "»+ι = 0{Αη-*ητ log An).
The theorem follows from Lemmas 14.1.3 and 14.1.12.
Proof of Theorem 14.1.2.
Note that 6 = 1 now. By the proof analogous to that of all the above
lemmas under the conditions of Theorem 14.1.2, we also have
Σ Tj - A\ = 0(nr (nr VBn) log3 An) a.s., (14.1.35)
j<n
where Sn-Efc<„«fc^n2·
Since
j<k j<k
and a? is non-increasing, we have
B„ = o(J>i)
k<n
by using the Abel transformation. Hence
ΣΤι-Αΐ = θ(ητ{ητνΒη)\οζ*Αη) a.s. (14.1.36)
j<n
The theorem follows from (14.1.36) and Lemma 14.1.3.

392
Chapter 14 Strong Approximations for Other Kinds
Remark 14.1.1. If k& = O(Ak) for some β > 0, \α^\ |, and |α^| =
0(к~в) for some r < θ < 1/2, then by the proof similar to that of Theorem
14.1.1, we have
S(A2n) - W{A2n) = 0(C^ log2 An) a.s.
where C„ = Efc<n Ы4-2г/*·
Let Sn(u) = Sfc<n λ/2<^θ8 2πη^α;, α; £ [0,1). Combining Corollary
14.1.1 with the results of the increments of a Wiener process, we can obtain
a.s. limiting behavior of increments of the sums Sn.
14.2 A class of Gaussian sequences
Let {Xn, η > 1} be a centered sequence of Gaussian random variables.
Under some conditions, including ε(ςΤ=™+ι Хк) = ησ<2 + 0(™1-ε) for
some ε > 0, σ2 > 0 and ЕХшХш+п = 0(n~2), Philipp and Stout (1975)
established strong approximations of partial sums
5(ί) = ΣΧ* t>0
k<t
by a Wiener process with order 0(ί1/2-λ), where 0 < λ < ~ Λ ||. By
applying the property of a symmetric matrix and the circle-plate theorem
about eigenvalues of a matrix, Shao (1985) proved an ideal result as follows:
Theorem 14.2.1. Suppose that there exist Ci > 0,г = 1,2,3, such
that for every η
m+n
E( ^2 Xk) > C\n uniformly in m (14.2.1)
k=m+l
EXl < C2 (14.2.2)
and
7(n) := sup \EXmXm+n\ < Czn-3'2-x (14.2.3)
m
for some λ > 0. Then without changing the distribution o/{5(t), t > 0}, we
can redefine the process {S(t),t > 0} on a richer probability space together
with a Wiener process {W{t),t > 0} such that
S(t) - W(bt) = 0{\og1'21) a.s. (14.2.4)

14.2 A class of Gaussian sequences
393
where
/M
bt = b[t] =at + ru at = ES2(t), rt = 0[Jjr k-1'2^). (14.2.5)
Corollary 14.2.1. If the condition (14-2.3) is strengthened by
7(71) < Cn-^logn)-1-6 (14.2.3х)
for some ε > 0, then we have
S(t) - W(at) = Otlog1/21) a.s. (14.2.4х)
Corollary 14.2.2. Suppose that {Xn,^ > 1} is a centered stationary
Gaussian sequence and (14-2.3') is satisfied, then
oo
σ2 := EX2 + 2 ^ ЕХгХк
к=2
converges absolutely. If σ2 > 0, then, assuming σ2 — 1, we have
S(t)-W(t) = 0(log1'2t) a.s.
The proof of the theorem needs the following lemmas.
Lemma 14.2.1. Let A be a real symmetric matrix of order η with
eigenvalues λι,···,λη. Denote λ = тах!<г<п |λ^|. Then for any row
vector С we have
\CAC'\ < ACCX (14.2.6)
where Cx is the transposed vector of С
Proof. We need only to prove that matrices XI — A and XI + A are
non-negative definite. By the well-known property of a matrix, there exists
a real orthogonal matrix U such that U'AU is a diagonal matrix Л with
the diagonal elements which is just equal to the eigenvalues of the matrix
A. Therefore we have
XI-A = U{XI-K)U'.
It is clear that the eigenvalues of XI — Л are all non-negative, so that the
eigenvalues of the real symmetric matrix XI — A are also non-negative.

394
Chapter 14 Strong Approximations for Other Kinds
Thus the matrix XI — A is non-negative definite, similarly, so is the matrix
XI + A. The lemma is proved.
Lemma 14.2.2. At least one of the following inequalities is satisfied
for eigenvalues of any matrix A = (α^)ηχη:
η
|А-од|< ]Г \aii\ ί = 1)···,^. (14.2.7)
This result, the so-called circle-plate theorem, is due to Gerschgorin
(cf. Franklin 1968, p. 161 Theorem 1).
Denote Tn = σ{Χ&, 1 < к < η}. Write
oo
Yn = ^(E(Xn+k\fn) - Е(Хп+к\Гп-г))
= Xn + un+1-un (14.2.8)
where щ = 0 and
oo
un = Σ Е(Хп+к\Гп-г), п = 2,3, ... (14.2.9)
k=0
It is clear that {Yn, Τη} is a martingale difference sequence. We shall prove
below that the series in (14.2.9) is convergent under the assumptions of
Theorem 14.2.1.
Lemma 14.2.3. // (Ц.2.1), (Ц.2.2) and (Ц.2.3) are satisfied, and
for any к > 1
oo
EX2k> £ \ЕХкХ^ + 1, (14.2.10)
then
K||2 = o(i).
Proof. Let A be the covariance matrix of (Χι,·-· ,Xj) and С =
(EXxXj+k, · · -tEXjXj+k). Then, by (5.22) in Philipp and Stout (1975),
we have
E{E2{Xj+k\^)) = CA-'C, (14.2.11)
for any j, к > 1. By condition (14.2.10) and Lemmas 14.2.1 and 14.2.2, we
obtain
СА'гС < CC'. (14.2.12)

14.2 A class of Gaussian sequences
395
Denote
unfc = £7(Xn+fc|^„_1), fc = 0,l,...;n=l,2,···. (14.2.13)
It follows from (14.2.11), (14.2.12) and (14.2.3) that
Eu2nk <П^(ЕХп+кХг)2
i=l
< Cl Σ(η + k ~ *)~3~2Х ^ Ci(k + 1)~2~2X- (14.2.14)
i=l
Thus we have
oo
1К1|2<Е11"»*Ь = о(1).
k=0
Proof of Theorem 14.2.1.
1) We first prove Theorem 14.2.1 under condition (14.2.10). By a
well-known property of a Gaussian sequence (cf. Ibragimov and Rozanov
1978, p.14), Yn, which is defined by (14.2.8), is a linear combination of
X1? · · · ,Xn, so that {Yn,n > 1} is also a Gaussian sequence. By (14.2.8)
{Y^,n>l}isa martingale difference sequence. Therefore Yn, η = 1,2,···
are independent. Then we can construct a Wiener process {W(t),t > 0}
from {Yn,n > 1}. Let bt = Ek<tEYk- We redefine
Yn = W(bn)-W(bn-1)i η = 1,2,.--
and from (14.2.8) we have
k<n k<n
Note that un, η = 1,2,···, are normal with uniformly bounded variances,
then we have
Un = OQog1'2 n) a.s.
i.e.
2^fc-Eyfc = °(1(«1/2n) a.s.
k<n k<n
or
S(t) - W(bt) = Oilog1/21) a.s. (14.2.15)

396
Chapter 14 Strong Approximations for Other Kinds
Furthermore
η
bn-an = 2E(J2 XkUn+i) + Eul+1
η oo
k=lj=l
π οο η
= o(EE-r(n+i-fc)) = o(s*-1/2-A).
fc=lj=l fc=l
Then Theorem 14.2.1 holds true under (14.2.10).
2) Now we prove Theorem 14.2.1 in general case. Put / = [312Cf/Cf],
define
X*m= Σ Хк m = l,2,.··, S; = ^X*k.
(m-l)l<k<ml k<t
Then {Xn} is also a Gaussian sequence satisfying (14.2.1), (14.2.2) and
(14.2.3). It follows from (14.2.1) that
EX*2 > Cxl. (14.2.16)
We prove that {X*} satisfies (14.2.10). In fact
Σ\Εχηχ;\
3=1
Эфгь
oo
= ΣΚ Σ ъ)( Σ *)Ι
j=\ (n-l)l<k<nl (j-l)l<i<jl
Зфгь
oo
<^Σ Σ Σ i*-*rs/2
j=l (n-l)l<k<nl (j-l)l<i<jl
Зфп
I oo
< 2C3 Σ Σ (^ + fc)~3/2
k=l m=0
I
< 6C3 Σ k~1'2 ^ 6Сз(1 + 2/1/2). (14.2.17)
By the definition of /, it is easy to verify that the right hand side of (14.2.17)
does not exceed C\l — 1, that is to say, {X^} satisfies (14.2.10).
For any fixed n, there exists an m such that (m — 1)1 < η < ml and
hence
|5n-5^|< max \Xk\. (14.2.18)
(m— l)l<k<ml

14.3 The non-negative additive functional of a Markov process
397
Note that {Xn,n > 1} is a Gaussian sequence with uniformly bounded
variances. Then we have
max \Xk\ = OilogV2 m) a.s. (14.2.19)
(m—l)l<k<ml
and
ml 2
an-am = EfjTXb) ~Ε(ΣΧ*Ϊ
k=l k=l
η ml ml 2
= -2Ε(Έ Σ XkXj)-E( ς xs)
k=lj=n+l j=n+l
oo
= θ(Σ7(*))=0(1), (14.2.20)
where a^ = E(Sm)2. It follows from 1), (14.2.18)-(14.2.20) that the proof
of Theorem 14.2.1 is completed.
14.3 The non-negative additive functional
of a Markov process
Let X = {Xt,t > 0} be a homogeneous, right continuous, strong-
Feller Markov process, which is defined on a probability space (Ω,^7, Ρ)
with values in a complete, σ-compact measurable metric space (£7,p,B),
satisfying the following conditions:
(i) for any xGiJ, t > 0 and open set U G B, the stationary transition
function of the Markov process X
p(t,x,U) >0; (14.3.1)
(ii) for every a £ Ε there exists a compact set К such that
Pa{Xt{u) G К for some t > 0} = 1, (14.3.2)
wherePa(-) = P(-|Xo = o).
Prom (i) and (ii) it follows that X is a recurrent strong Markov process.
For every U G B, put
ί inf{£, t > 0, Χί(α;) G Ϊ7}, if the set is non-empty,
[ oo otherwise.

398
Chapter 14 Strong Approximations for Other Kinds
Define the exit distribution
hu{a, S) = Ρα{ΧΤυ{ω) e S,tv < 00} for a G E, S G B.
Denote
Λ*7(*) = LhU(x,dy)f(y) for /(*) G В(ЁГ),
where B(U) is the set of all bounded measurable function on U.
Let H= {K, L} be the collection of all set-pairs satisfying the following
conditions:
(Hi) К and L are closed subsets of Ε and have an interior point at
least,
(H2) К and/or L are compact,
(Щ) (i) К С E\L, E\L is a connected open set, or
(ii) L С Е\К, E\K is a connected open set.
For any given set-pair (K, L) G Η, χ £ K, put
T*(*, t/) = / /iL(z, <*у)Л*(у, Е7), (14.3.3)
JE
where U G B, U С К. By recurrence TK(x, U) is a one-step transition
function. Define the transformation
T*/(a;)= { TK{xbdy)f{y) (14.3.4)
for ж G ii, / G B(tf).
Without loss of generality, assume that К is compact, and (i) of (Щ) is
satisfied. Then for TK there exists a unique invariant probability measure
μ on K. Moreover, suppose that X satisfies
(iii) TK{B(K)} С C(K) := {all continuous function on K}.
Let r be a stopping time of the process X, put Ωτ = {τ(ω) < oo}.
Then
ЛГТ := {Л : А С Ωτ, Vt > 0, Α Π (r < t) G M}
is a σ-field of Ωτ, where Mt = σ{Χδ,0 < 5 < t}. Let ЛГ* be a σ-field of Ω
which was introduced in Dynkin (1963 (3, 3.5)). Define the shift operators
9t from M* to M* and the shift operator θτ from M* to Ω,ΤΜ* satisfying
ΘΤΑ = (J {Μ, τ(ω) = 1}СПТ АеМ*, (14.3.5)
where the operations of union, intersection and complement are keeped by
the operator 0T, and we have
er{xt er} = {xt+T eг} гg b. (14.3.6)

14.3 The non-negative additive functional of a Markov process
399
For given (K, L) G H, we define random functions on Ω as follows:
П =t1(K,L,u)
finf{£, t > 0, Xt(u) G K} if the set is non-empty,
oo otherwise;
ση = an(K,L,u)
finf{£, t > rn,Xt(u;) G L} if the set is non-empty,
oo otherwise;
Tn+i = rn+i(K,L,u;)
f inf{t,t > an,Xt{uj) G K} if the set is non-empty,
oo otherwise,
for η > 1. By Doob (1953), rn, ση (η > 1) are the almost sure finite
stopping times of the process X. Denote £η(ω) — ΧΤη(ω). Its transition
function TK(x, U) satisfies the Doeblin condition and there exists a unique
ergodic set, non-periodic. Moreover TK(x,U) satisfies
\(TK)n(x, U) - μψ)\ < C6n, (14.3.7)
where 0 < 6 < 1, С is a constant. Put
P(B) = ( Ρα(Β)μ(άα) Be Т. (14.3.8)
JE
Then Ρμ is a probability measure on (Ω,.77), and Pa(B) — 1 implies
Ρμ(Β) = 1. Put
Εμί(·) = Ι ί(')Ρμ(άω).
Jn
At last, let φ = {φΐ(ω)} be a non-negative, strongly measurable,
homogeneous additive functional on Ω, i.e. {(/>*, 0 < 5 < t} is a family of real
valued functions satisfying the following condtions:
(φι) For any s < £, Φ1(ω) is non-negative ~tft-measurable, where ~tft is
the completion of Aft in probability space (Ω,^7, Ρ),
(^2) for any ω G Ω and 5 < t < и, ф8и = Ф1 + фьи,
(фз) for any ω G Ω and /ι > 0, s < t, Вкф\ = <#+£,
(φι) for any 0 < и < г>, the bivariate function φ®(ω) of point (ί,ω) is
B[u,v] x ЛС-measurable on [щ ν] χ Ω, where B[UjU] is the σ-field of interval
[u,v], J7^ is the completion of Af„ — a(Xt,u <t<v).
It is easy to check that
θσιτη = rn+i - σχ. (14.3.9)

400
Chapter 14 Strong Approximations for Other Kinds
Denote
Уп — Φτ\ι+1'> ζη = 7"n+l — Tn-
Then
θσιυη = Уп+ι, θτιζη = ζη+χ. (14.3.10)
Lemma 14.3.1. {yn, n > 1} is a strictly stationary φ-mixing sequence
of random variables with Y^=\ (^1/2(η) < oo.
Particularly, {zn,n > 1} and {wn = yn + czn,n > 1} are a/so 2/ie
strictly stationary φ-mixing sequences.
Proof. In order to prove {yn} is a strictly stationary sequence,
it needs only to show that θσι on Ρμ is an operator preserving measure.
Indeed, from (14.3.8) and Dynkin (1963 Theorem 3.11), for any В G λί0 :=
0"{-X"t5 t > 0} we have
Ρμ(θσιΒ) = ί ΡΧσι (Β)Ρμ(άχ) = Ρμ{Β).
In order to prove {yn} is (^-mixing, we first show the following two
facts:
(i) a{yi,---,ym_i} С σ{ξι,···,ξ„ι} CJVTm,
(U) <7{Ут+к-11Ут+к,-'} С a{fm+fc,fm+fc+b*· '}·
Since ΜΤΎη_λ С Λ/"Ττη and £m = XTm is Λ/"Ττη-measurable, £fc, 1 < /с < m are
also Λ/^-measurable. Therefore
σ{6," ··,&*} CJVTm.
Thus (i) holds true, if we can prove
a{yn GA, AGBR}C σ{£η+χ G Γ, Γ G В} (14.3.11)
where Br is Borel σ-field of i?+. Put
Л = {Л: (yn G Л) G σ{^η+ι G Γ,Γ G В}, Л G Яд},
it is easy to verify that Λ is a λ-system and contains a π-system
Π = {[0,t] : (yn G [0,t)) G σ{ξη+1 G Γ,Γ G B}}.
Then σ(Π) С Λ that implies (14.3.11). The proof of (ii) is similar.
Thus for any A G o{yk, 1 < к < m — 1}, Be o{yk, к > m + η — 1},
by the strong Markov property and (14.3.7) we have
Ρμ(ΑΒ) = Ям(ВДЛ)/(Я)|Л/;т))
= Εμ{Ι{Α)Εμ{Ι{Β)\ΜΤπί)) = адЛ)Я„(/(В)|£т))

14.3 The non-negative additive functional of a Markov process
401
and
\Ρμ(ΑΒ) - Ρμ(Α)Ρμ(Β)\
= \Εμ(Ι(Α)Εμ(Ι(Β)\ξη)) - ΕμΙ{Α)ΕμΙ{Β)\
= \Εμ{ΐ(Α)ΙΕΕμ(Ι(Β)\ξτη+η=η)[(ΤΚηξτη,άη)-μ(άη)]}\
< Εμ{ΐ(Α) J ЗД(В)|£т+п = n)Vn(U,dn)}
< E„I(A)Vn(U, E) < Ρμ(Α)Οδη, (14.3.12)
where Vn(U, A) = \(Тк)п(£т, А) - μ(Α)\. Taking ψ{η) = C6n(0 < δ < 1)
we obtain {yn} is a y?-mixing sequence with Ση^=ι φχΙ2(η) < oo.
Particularly, letting <f>st — t — s, we have yn = zn, so that the result of
{zn} holds true. Moreover, from (i) and (ii)
°{wk, l<fc<m-l} С σ{&, 1 < к < m} С Л/"Тт,
a{wk, к > т + п — 1}С a{£k, к> т + п}.
{wn} is also a strictly stationary ψ-mixing sequence. Lemma 14.3.1 is
proved.
Lemma 14.3.2. Let ф\ and ΦΙ be the non-negative, strongly
measurable, homogeneous additive functionals of X with the finite αψ = Ε^ι, αψ
= Εμψ% ф 0. Then
(a)
Pa{ lim (fi/tfi = αΦ/αΛ = p{ lim φ°/ψ° = αφ/αΔ = 1 (14.3.13)
where
P(A) = ( Pa(A)P0(da), (14.3.14)
JE
Pq is an initial distribution.
(b) Particularly, letting l(t) to be a positive integer-valued random
variable such that
4{t) < t < 7i(t)+1 (14.3.15)
and az = Εμζη, we have
p»te m=°<}=Fte щ="■}='· <14·316)
*·{&?->}-'tea3?1-'J-1· <14·3·17'

402
Chapter 14 Strong Approximations for Other Kinds
Proof. By the strict stationarity and (^-mixing property of {yn}5
{yn} is an ergodic sequence and its invariant σ-field U is trivial. By (14.3.8)
for any A £ U
Pa(A) = Ρμ(Α) = 0orl.
Hence we have
ι n
Ρμ\ lim - X] yk = αφ\ = 1.
k=l
And since lim sup — Σ£=ι Ук and liminf — Σ£=ι Ук are ZY-measurable, we
have
ι n
Pa{ lim - V yfc = «Л = 1. (14.3.18)
k=l
Obviously l(t) —> oo(£ —> oo). Write
*? = <+ Σ> + *?(°· (14-3.19)
fc=l
Since τχ < oo a.s., for any given ε > 0 and large t we have
P„{</i(t) > ε} = 0. (14.3.20)
By the non-negativity of φ\, for large t we also have
Pa{itm/Kt) >e} = Р«{^$+1//(«) > ε} - 0. (14.3.21)
Combining (14.3.18)-(14.3.21), we get
Pa{ lim 0?//(t) - αφ} = 1. (14.3.22)
Similarly we have
Pa\ lim ψ°/1(ί) = α^} = 1. (14.3.23)
Putting (14.3.22), (14.3.23) and (14.3.14) together yields (14.3.13).
Note that (14.3.16) is a special case of (14.3.22) with φ\ = t - s.
Moreover
rl{t) _ η l(t) - 1 1 ^
Prom η < oo a.s., (14.3.18) and (14.3.16), (14.3.17) holds true.
Now let us consider the process S(t) generated by the additive
functional φ® as follows:
S(t) = <fi- Mt.

14.3 The non-negative additive functional of a Markov process
403
Put
ΨΙ = t - s, Μ = αφ/αψ.
Lu (1986) proved the following theorem
Theorem 14.3.1. Let ΦΙ be the non-negative, strongly measurable,
homogeneous additive functional ofK, wn = yn — Mzn. Suppose that for some
0 < δ < 2,
Εμυ2η+δ < oo, Εμζ2η+δ < oo.
Then Euwl+6 < oo and
^μ^η
a2w = Εμν)\ + 2 ]Γ Я^г^
fc=2
converges absolutely. Without loss of generality assume that σ^/αψ = 1,
2/ien without changing the distribution of S(t), we can redefine the process
{S(t),t > 0} on a richer probability space together with a Wiener process
{W(£), t > 0} such that for any given ε > 0 a.s.
(o(t*+*+£) if 0<<5<2,
S(t) - W(t) = { ), ' 9 ч
(o(t4(logi)4+ej »/ δ = 2.
Proof. We can write
l(t)-i
5(t) = $ - Mt = < + £ ti;* + ^W - M(t - rl(t) + n).
fc=l
Denote Z(£) = Σ^_ι ^Ь we have
5(t) - Z(t) = ф°Т1 + фУ - M(t - rl{t) + n). (14.3.24)
We prove that the right hand side of (14.3.24) a.s. equals to
O^iloglogi)1^2"^)). By Lemma 14.3.1 {yn} is a strictly stationary ψ-
mixing sequence with φ(η) = Се~вп, θ > 0. Moreover {yn + '' ,n > 1} is
also a strictly stationary <£>-mixing sequence with
д(у(2+*)/2)2 = ду2+6 < ^
Therefore by the law of iterated logarithm we have
£(νΐ+6)/2 - ЕуГ6У2) = 0((nloglogn)1^) a.s.
fc=l

404
Chapter 14 Strong Approximations for Other Kinds
From the non-negativity of additive functional ΦΙ and Lemma 14.3.2 we
obtain
<ftm < Vi(t) = O((i(i)loglog/(0)1/(2+5)) = 0((t log log i)1/^)) a.s.
Similarly we also have
t - nit) < 4t) = C((iloglogi)1/(2+i)) a.s. (14.3.25)
Obviously <$0 = O(l), Μτχ = O(l). Thus we obtain
S{t) - Z(t) = Odtloglogty/V+V) a.s.
For the strictly stationary ^-mixing sequence {wn}, from Theorems 9.1.1
and 9.1.2 we have a.s.
(o(t^+£) ifO<<5<2
J2wk-W(nal)={ K J (14.3.26)
Now we write
Z(t) - W(t)
= (£>*- W((l(t) - l)d)) + (W((l(t) - 1)σ*) - W(tj)
=:/i+/2. (14.3.27)
From Lemma 14.3.2 and (14.3.26) we have a.s.
( o(l(t)^s+£) = 0(t^s+£) if 0 < δ < 2
'ι = 1 / ι θ ч ι 9 (14.3.28)
[ o(/(t)4(log/(t))4+ej = 0(*4(l0g*)4+e) if 5 = 2.
On the other hand, by the law of iterated logarithm for the strictly
stationary <£>-mixing sequence {zn} we have
η
Y^zk~ ηαψ = <3((n log log n)1/2) a.s.
Hence
rl(t) - 1(ί)αφ = <9((Ζ(ί) log log Ζ(ί))1/2) = 0((iloglogi)1/2) a.s.,
so that
t - Щ)аф = t - l{t)a2w = 0((iloglogi)1/2) a.s.

14.3 The non-negative additive functional of a Markov process
405
By theorem 1.2.2 of Сsorgo and Revesz (1981) we have
h = W((l(t) - 1)αφ) - W(t) = Odtloglogt)1/4) a.s.
Combining it with (14.3.25)-(14.3.27) we have a.s.
( o(t^+£) if 0 < 6 < 2
S(t) - W(t) ={ ), } 9 N
[o(t4(logi)4+ej if 6 = 2.
Theorem 14.3.1 is proved.
Remark 14.3.1. From Theorem 14.3.1 we can give a weak invariance
principle and a law of iterated logarithm for the additive functional of a
Markov process.

Appendix Slowly Varying Function
Definition Al. A positive and measurable function R(x) on [Л, oo)
for some A > 0 is called regularly varying at infinite point with an exponent
a, if for any а > 0
Um^ R(ax)/R(x) = aa. (Al)
Rewrite a regularly varying function R(x) with an exponent a as
R(x) =xaL(x). (A2)
Then, by (Al), we have
lim L(ax)/L(x) = 1.
χ—►oo
Definition A2. A regularly varying function L(x) with the exponent
a = 0 is called a slowly varying function.
We shall list some main propositions on a slowly varying function. For
their proof, we refer to Seneta (1976) and Ibragimov and Linnik (1971).
There are Karamata's representation theorem for a slowly varying
function.
Theorem Al. Let L(x) be α slowly varying function defined on [Л, оо), А
0. Then there exists a positive В > A such that for any χ > В
L(x) = ехр{т7(ж) + / -γ dt}>
where η(χ) is a bounded measurable function on [5, oo) with^x) —► c(\c\ <
oo) as χ —> oo and ε(χ) is a continuous function on [S, oo) with ε(χ) —► 0
as χ —> oo.
Using this representation theorem, we can derive many useful
properties. In the sequel, we always assume that L(#), L\{x), L,2(x) are slowly
varying functions.

Appendix Slowly Varying Function 407
Property Al. For any a > 0.
lim L(x + a)/L(x) = 1.
χ—>οο
Property A2. For any ε > 0,
lim x£L(x) = oo, lim x~£L(x) = 0.
x—>oo x—>oo
Property A3. 8ир2;ь<£<2^+1 L(t)/L(2k) —► 1, as A: —► oo.
Property A4. Let α — α(#) —> 0 as ж —> oo. Then for any ε > 0,
lim α£ Λ / = lim a£ r / \ = 0.
ж-юо L(x) ж-юо L(ax)
Property A5. (logL(#))/log# —> 0 as ж —> oo.
Property A6. For any real number a, La(#), Li(#)Z/2(#) and Li(x)+
L2(x) all are slowly varying. Moreover, if Z^Oc) —> oo as ж —> oo, then
Li(L2(x)) also is slowly varying.
Property A7. Define L{x) and L(x) by
xJL(x) = sup tJL(t),
B<t<x
χΊΣ(χ) = inf Г L(t),
where 7 > 0 is an arbitrary constant. Then L ~ L and L~L.
Remark Al. As a consequence of Property A7, xJL(x) is equal
asymptotically to a non-decreasing regularly varying function with the
same exponent 7.
Property A8. For R\(x) = xJLi(x), 7 > 0, there exists a regularly
varying function B.2(x) = x1/1L,2(x) such that
i?i(i?2(#)) ~ ж, R2(Ri(x)) ~ χ as ж —> oo.
R2(x) here is defined asymptotically uniquely, i.e. if the above relations
hold true with i?3 instead of i?2 and Яз(х) —> oo as ж —> oo, then Дз(ж) ~
x1/7L2(^)·

408
Appendix Slowly Varying Function
Property A9. Let L(x) is a positive slowly varying function on
[A, oo). Assume that R(x) = χΊΣ{χ) is non-decreasing on [A, oo] for some
7 > 0. For χ > R(A) let
R*(x) = inf{y : у G [A, oo), R{y) > x}.
Then R*(x) = xl^L*(x), where L*(x) is a slowly varying function and
R*(x) is an inverse function of R(x) with the meaning in Property A8.

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Index
α-mixing 1.1, 6.2
(a,/3)-mixing 1.2
absolutely regular 1.1, 6.2
additive functional of Markov
process 14.3
^-mixing 1.1
Bernstein inequality 11.1, 12.4
Berry-Esseen inequality 7.1
~ for U-statistic 13.1
bisection lemma 6.2
bounds of covariances 1.2
bounds of the variances οξ
partial sums 2.1
Central limit theorem (CLT) 3
~ for α-mixing random field 6.1
~ for /9-mixing random field 6.1
~ necessary and sufficient
condition 3.1
~ sufficient condition 3.2
~ with infinite variance 3.3
complete convergence 8.3-8.5
~ for a-mixing sequence 8.5
~ for φ-mixing 8.3
~ for /9-mixing 8.4
density function 13.3
kernel estimates of ~ 13.3
nearest neighbor estimates of ~
13.3
empirical process 12
error variance in linear model 13.2
exponential inequality 10.1
Gaussian sequence 14.2
Ibragimov-Linnik-Iosifescu
conjecture 5.2
increments of partial sum 10
~ of φ-mixing sequence 10.1
~~ with moment generation
function 10.1
~~ with finite p-order moment
10.1
inequality
~ of tail probability 2.2
~ of the moments of partial sums
2.2
~ of the moments of maximum
partial sums 2.2
lacunary trigonometric series 14.1
law of the iterated logarithm 9.2
Levy-Prohorov distance 7.2
metric entropy condition 6.2
moduli of continuity of empirical
process 12.4
~ with α-mixing sample 12.4
~ with (^-mixing sample 12.4

426
Index
nonuniformly (^-mixing 6.2
Ottaviani inequality 2.2
(^-mixing 1.1, 6.2
c^*-mixing 13.1
^-mixing 1.1
Prohorov problem 8.6
~ for additive functional of
Markov process 14.3
~ for empirical process 12.3
~ for empirical process 12.3
~ for error variance estimations
in linear model 13.2
~ for Gaussian sequence 14.2
~ for lacunary trigonometric
series 14.1
strong law of large numbers 8.2
~ for U-statistic 13.1
p-mixing 1.1, 6.2
random field 6.1, 6.2
α-mixing ~ 6.1, 6.2
a*-mixing ~ 6.1
p-mixing ~ 6.2
absolutely regular ~ 6.2
nonuniformly (^-mixing ~ 6.2
symmetric c^-mixing ~ 6.2
rate of convergence 7
~ in distribution 7.1
~~ for α-mixing sequence 7.1
~~ for p-mixing sequence 7.1
rate of weak convergence 7.2
~ for a-mixing sequence 7.2
~ for (^-mixing sequence 7.2
~ for absolutely regular sequence
7.2
*-mixing 1.1
set-indexed partial sum process 6.2, 6.3
set indexed empirical process 12.3
slice 6.2
slowly varying function 2.1,App.
strong approximation 9
~ for α-mixing random
field 11.2
~ for (^-mixing random
field 11.1
~ for U-statistic 13.2
thickness 6.2
totally bounded 6.2
~ with inclusion 6.2
U-statistics 13.1
uniformly integrablity 2.1
uniform empirical process 12.1
uniformly mixing 1.1
unrestricted p-mixing
sequence 6.1
Vapnik-Cervonenkis class 6.2
von-Mises statistics 13.1
weak convergence 3, 4, 5
~ for a-mixing sequence 3
~ for p-mixing sequence 4
~ of empirical process 12.1
weak invariance principle
sufficient condition for ~ 3.2
~ when variance is finite 4.1, 5.1
~ when variance is infinite 3.3, 4.4
~ for error variance estimations
in linear model 13.2
~ for U-statistics 13.1
weak law of large number 8.1

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