random walk in random and non-random environments - by pal revesz
TRANSCRIPT
RANDOM WALK
IN RANDOM AND
NON-RANDOM ENVIRONMENTS
Pal ReveszTechnical University of Vienna, Austria
Technical University of Budapest, Hungary
World ScientificSingapore • New Jersey • London • Hong Kong
NON-ACTIVATEDVERSIONwww.avs4you.com
Published by
World Scientific Publishing Co. Pte. Ltd.,
P O Box 128, Farrer Road, Singapore 9128
USA office: 687 Hartwell Street, Teaneck, NJ 07666
UK office: 73 Lynton Mead, Totteridge, London N20 8DH
Library of Congress Cataloging-in-Publication Data
Revesz, Pal.
Random Walk in random and non-random environments/Pal Revesz.
p. cm.
Includes bibliographical references (P. ) and indexes.
ISBN 9810202377
1. Random walks (Mathematics) I. Title.
QA274.73.R48 1990
519.2'82-dc20 90-37709
CIP
Copyright © 1990 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproducedin any form or by any means, electronic or mechanical, including photo-photocopying, recording or any information storage and retrieval system now
known or to be invented, without written permission from the Publisher.
Printed in Singapore by JBW Printers and Binders Pte. Ltd.
NON-ACTIVATEDVERSIONwww.avs4you.com
Preface
"I did not know that it was so dangerous to drink a beer with you. You write
a book with those you drink a beer with," said Professor Willem Van Zwet,referring to the preface of the book Csorgo and I wrote A981) where it was told
that the idea of that book was born in an inn in London over a beer. In spiteof this danger Willem was brave enough to invite me to Leiden in 1984 for a
semester and to drink quite a few beers with me there. In fact I gave a seminar
in Leiden, and the handout of that seminar can be considered as the very first
version of this book. I am indebted to Willem and to the Department of Leiden
for a very pleasant time and a number of useful discussions.
I wrote this book in 1987-89 in Vienna (Technical University) partly sup-
supported by Fonds zur Forderung der Wissenschaftlichen Forschung, Project Nr.
P6076. During these years I had very strong contact with the Mathematical
Institute of Budapest. I am especially indebted to Professors E. Csaki and A.
Foldes for long conversations which have a great influence on the subject of this
book. The reader will meet quite often with the name of P. Erdos, but his role in
this book is even greater. Especially most results of Part II are fully or partly due
to him, but he had a significant influence even on those results that appearedunder my name only.
Last but not least, I have to mention the name of M. Csorgo, with whom I
wrote about 30 joint papers in the last 15 years, some of them strongly connected
with the subject of this book.
Vienna, 1989. P. Revesz
Technical University of Vienna
Wiedner Hauptstrasse 8-10/107A-1040 Vienna
Austria
NON-ACTIVATEDVERSIONwww.avs4you.com
Contents
Preface v
Introduction xiii
I. SIMPLE SYMMETRIC RANDOM WALK IN Zl
Notations and abbreviations 3
1 Introduction of Part I 9
1.1 Random walk 9
1.2 Dyadic expansion 10
1.3 Rademacher functions 11
1.4 Coin tossing 11
1.5 The language of the probabilist 11
2 Distributions 13
2.1 Exact distributions 13
2.2 Limit distributions 19
3 Recurrence and the Zero-One Law 23
3.1 Recurrence 23
3.2 The Zero-One Law 25
4 From the Strong Law of Large Numbers to the Law of
Iterated Logarithm 27
4.1 Borel - Cantelli Lemma and Markov inequality 27
4.2 The strong law of large numbers 28
4.3 Between the Strong Law of Large Numbers and the Law of
Iterated Logarithm 30
4.4 The LIL of Khinchine 31
vii
NON-ACTIVATEDVERSIONwww.avs4you.com
viii CONTENTS
5 Levy Classes 33
5.1 Definitions 33
5.2 EFKP LIL 35
5.3 The laws of Chung and Hirsch 39
5.4 When will Sn be very big? 39
5.5 A theorem of Csaki 41
6 Wiener process and Invariance Principle 47
6.1 Two lemmas 47
6.2 Definition of the Wiener process 48
6.3 Invariance Principle 52
7 Increments 55
7.1 Long head-runs 55
7.2 The increments of a Wiener process 63
7.3 The increments of Ss 73
8 Strassen type theorems 79
8.1 The theorem of Strassen 79
8.2 Strassen theorems for increments 86
8.3 The rate of convergence in Strassen's theorems 88
8.4 A theorem of Wichura 90
9 Distribution of the local time 93
9.1 Exact distributions 93
9.2 Limit distributions 99
9.3 Definition and distribution of the local time of a Wiener process 100
10 Local time and Invariance Principle 105
10.1 An Invariance Principle 105
10.2 A theorem of Levy 107
11 Strong theorems of the local time 113
11.1 Strong theorems for ?{x,n) and f(n) 113
11.2 Increments of ri(x,t) 115
11.3 Increments of ?{x,n) 118
11.4 Strassen type theorems 120
11.5 Stability 122
11.6 Favourite points . 129
11.7 Rarely visited points 132
NON-ACTIVATEDVERSIONwww.avs4you.com
CONTENTS ix
12 An embedding theorem 133
12.1 On the Wiener sheet 133
12.2 The theorem .134
12.3 Applications 137
13 Excursions 141
13.1 On the distribution of the zeros of a random walk 141
13.2 Local time and the number of long excursions
(Mesure du voisinage) 147
13.3 The local time of high excursions 152
13.4 How many times can a random walk reach its maximum? .... 157
14 A few further results 161
14.1 On the location of the maximum of a random walk 161
14.2 On the location of the last zero 165
14.3 The Ornstein - Uhlenbeck process and a theorem of
Darling and Erdos 169
14.4 A discrete version of the Ito formula 173
15 Summary of Part I 177
II. SIMPLE SYMMETRIC RANDOM WALK IN Zd
Notations 181
16 Recurrence theorem 183
17 Wiener process and Invariance Principle 189
18 The Law of Iterated Logarithm 193
19 Local time 197
19.1 f@, n) in Z2 197
19.2 f (n) in Zd 204
19.3 A few further results 206
20 The range 207
20.1 The range of Sn 207
20.2 Wiener sausage 210
NON-ACTIVATEDVERSIONwww.avs4you.com
x CONTENTS
21 Selfcrossing 213
22 Large covered balls 217
22.1 Completely covered discs in Z2 217
22.2 Discs covered with positive density 233
22.3 Completely covered balls in Zd 241
22.4 Once more on Z2 248
23 Speed of escape 249
24 A few further problems 255
24.1 On the Dirichlet problem 255
24.2 DLA model 258
24.3 Percolation 260
III. RANDOM WALK IN RANDOM ENVIRONMENT
Notations 263
25 Introduction 265
26 In the first six days 269
27 After the sixth day 273
27.1 The recurrence theorem of Solomon 273
27.2 Guess how far the particle is going away in an RE 275
27.3 A prediction of the Lord 277
27.4 A prediction of the physicist 287
28 What can a physicist say about the local time f@,n)? 291
28.1 Two further lemmas on the environment 291
28.2 On the local time f@, n) 292
29 On the favourite value of the RWIRE 297
30 A few further problems 305
30.1 Two theorems of Golosov 305
30.2 Non-nearest-neighbour random walk 307
30.3 RWIRE in Zd 308
30.4 Non-independent environments 310
NON-ACTIVATEDVERSIONwww.avs4you.com
CONTENTS xi
30.5 Random walk in random scenery 310
30.6 Reinforced random walk 311
References 313
Author Index 327
Subject Index 331
NON-ACTIVATEDVERSIONwww.avs4you.com
Introduction
The first examinee is saying: Sir, I did not have time enough to study everythingbut I learned very carefully the first chapter of your handout.
Very good -
says the professor -
you will be a great specialist. You know what
a specialist is. A specialist knows more and more about less and less. Finally he
knows everything about nothing.The second examinee is saying: Sir, I did not have enough time but I read your
handout without taking care of the details. Very good - answers the professor -
you will be a great polymath. You know what a polymath is. A polymath knows
less and less about more and more. Finally he knows nothing about everything.
Recalling this old joke and realizing that the biggest part of this book is
devoted to the study of the properties of the simple symmetric random walk (orequivalently, coin tossing) the reader might say that this is a book for specialistswritten by a specialist. The most trivial plea of the author is to say that this
book does not tell everything about coin tossing and even the author does not
know everything about it. Seriously speaking I wish to explain my reasons for
writing such a book.
You know that the first probabilists (Bernoulli, Pascal, etc.) investigated the
properties of coin tossing sequences and other simple games only. Later on the
progress of the probability theory went into two different directions:
(i) to find newer and deeper properties of the coin tossing sequence,
(ii) to generalize the results known for a coin tossing sequence to more com-
complicated sequences or processes.
Nowadays the second direction is much more popular than the first one. In
spite of this fact this book mostly follows direction (i). I hope that
(a) using the advantage of the simple situation coming from concentrating on
coin tossing sequences, the reader becomes familiar with the problems, results
and partly the methods of proof of probability theory, especially those of the
limit theorems, without suffering too much from technical tools and difficulties,
(b) since the random walk (especially in Zd) is the simplest mathematical
model of the Brownian motion, the reader can find a simple way to the problems
xiii
NON-ACTIVATEDVERSIONwww.avs4you.com
xiv INTRODUCTION
(at least to the classical problems) of statistical physics,(c) since it is nearly impossible to give a more or less complete picture of the
properties of the random walk without studying the analogous properties of the
Wiener process, the reader can find a simple way to the study of the stochastic
processes and should learn that it is impossible to go deeply in direction (i)without going a bit in direction (ii),
(d) any reader having any degree in math can understand the book, and
reading the book can get an overall picture about random phenomena, and the
readers having some knowledge in probability can get a better overview of the
recent problems and results of this part of the probability theory,
(e) some parts of this book can be used in any introductory or advanced
probability course.
The main aim of this book is to collect and compare the results -mostly strongtheorems - which describe the properties of a simple symmetric random walk.
The proofs are not always presented. In some cases more proofs are given, in
some cases none. The proofs are omitted when they can be obtained by routine
methods and when they are too long and too technical. In both cases the reader
can find the exact reference to the location of the (or of a) proof.
NON-ACTIVATEDVERSIONwww.avs4you.com
"Four legs good, two legs better."
A modified version of f.he
Animal Farm's Constitution.
"Two logs good, p logs better."
The original Constitution
of mathematicians.
I. SIMPLE SYMMETRIC
RANDOM WALK IN lNON-ACTIVATEDVERSION
www.avs4you.com
Notations and abbreviations
Notations
General notations
1. Xi,X2,... is a sequence of independent, identically distributed random
variables with
2. So = O,Sn = S{n) = Xl + X2 + --- + Xn (n = 1,2,...).{Sn} is the (simple symmetric) random walk.
3. M+ = M+(n) = maxSk,n V '0<k<n
'
M~ = M~(n) = — min S*,n V '0<k<n
*'
Mn = M(n) = max |5jt| = max(M^",M~),0<fc<n
4. {W(t);t > 0} is a Wiener process (cf. Section 6.1).
5. m+{t) = supW(s),0<»<t
m-{t) = - inf W(s),0<s<t
V '
m(t) = sup \W(s)\ = max(m+{t),m-{t)) {t > 0),0<»<t
m*(t) = m+(t)+m-{t),= m+{t)-W{t).
6. 6n = 6(n) = Bnloglogn)-1/2,In = l{n, a) = 12a Hog - + log log n) )
7. [x] is the largest integer less than or equal to x.
NON-ACTIVATEDVERSIONwww.avs4you.com
I. SIMPLE SYMMETRIC RANDOM WALK IN Zl
8. f(n) > g(n) «-> g(n) = o(f(n)) «-> Jun-—¦ = oo.
9. g(n) = O(f(n)) <-+ 0 < Iiminf44 < Iimsup44 <
10. /(n) « g(n) <-> lim —)—r- = 1.
11. Sometimes we use the notation f(n) ~ g(n) without any exact mathemati-
mathematical meaning, just saying that f(n) and g(n) are close to each other in some
sense.
12. $(z) =.— / e~u /2du is the standard normal distribution function.
13. N eN{m,a) ^P{a~l(N - m) <x} = $(z).
14. #{...} =| {...} | is the cardinality of the set in the bracket.
15. Rd resp. Zd is the d-dimensional Euclidean space resp. its integer grid.
16. B = Bd is the set of Borel-measurable sets of Rd.
17. A() is the Lebesgue measure on Rd.
18. logp (p = 1,2,...) is p-th iterated of log and lg resp. lgp is the logarithmresp. p-th iterated of the logarithm of base 2.
19. Let {Un} and {Vn} be two sequences of random variables.
{Un,n= 1,2,...} = {Vnn = 1,2,...} if the finite dimensional distributions
of {Un} are equal to the corresponding finite dimensional distributions of
Notations to the increments
1. h{n,a) = max (Sk+a - Sk),0<k<n-a
2. h{n,a) = QnnxJSk+a- Sk\,
3. h(n, a) = max
4. lAn,a) = max max|Sjt+,- — Sid,
NON-ACTIVATEDVERSIONwww.avs4you.com
NOTATIONS AND ABBREVIATIONS 5
5. h{n,a) = min max I S*+,-— S* |,0<k<n~a0<j<a
6. J1{t,a)= sup {W{s + a) -W{s)),0<S<t-a
7.J2{t,a)= sup \W(s + a)-W{s)\,0<s<t-a
8. J3(t,a)= sup sup (W(s+u)- W(s)),0<t<t-a0<u<a
9. J4{t, a) = sup sup \W(s + u) - W(s)\,0<3<t-a0<u<a
10. JM,a)= inf sup \W(s + u) - W(s)\,0<«<t-a0<u<a
11. Zn < n is the largest integer for which h(n, Zn) = Zn, i.e. Zn is the lengthof the longest run of pure heads in n Bernoulli trials.
Notations to the Strassen-type theorems
1. Sn(x) = bn Unz] +(x- I~H X[nx]+1\ @ < x < 1),
2. iut(x) = btW(tx) @<x<l;t>0),
3. C@,1) is the set of continuous functions defined on the interval [0,1],
4. S@,1) is the Strassen's class, containing those functions /(•) ? C@,1) for
which /@) = 0 and ^{f'{x))Hx < 1.
Notations to the local time
1. ?(x, n) = #{k : 0 < k < n, Sk = x} (x = 0, ±1, ±2,...; n = 1,2,...) is
the local time of the random walk {Sk}. For any A C Z1 we define the
occupation time E(A, n) = Ezex f (x>n)-
2. 77B:,*) (-cx> < x < +oo;t > 0) is the local time of W(-) (cf. Section 9.3).
3. H(A,t) = \{s : 0 < 5 < t,W(s) 6 A} (A c /21 is a Borel set, t > 0) is the
occupation time of W(-) (cf. Section 9.3).
4. Consider those values of k for which S* = 0. Let these values in increasingorder be 0 = p0 < pl < p2 < ..., i.e. pl = min{A; : k > 0, Sk = 0},p2 =
mm{k : k > Pl, Sk = 0},..., pn = m\n{k : k > pn-USk = 0},....
NON-ACTIVATEDVERSIONwww.avs4you.com
I. SIMPLE SYMMETRIC RANDOM WALK IN Z1
5. Similarly for any x = 0, ±1,±2,... consider those values of k for which
Sk = x. Let these values in increasing order be 0 < Pi(x) < p2(x) < ...
i.e. Pi(x) = min{A; : k > 0, Sk = x},p2(x) = min{A; : k > pi(x),Sk =
x}-> • • • ,Pn{x) = min{A; : k > pn-i(x),Sk = x] ... Clearly p,@) = /v In case
of a Wiener process define p*u = inf{t : t > O,ij(O,t) > u}.
6. ^(n) = max, ?(x,n).
7. ri(t) =supxV(x;t).
8. The random sequences
Si,... ,SPi}, E2 = \SP1,
are called the first, second, ... excursions (away from 0) of the random
walk {Sk}.
9. The random sequences
Ei(x) =
E2(x) =
are called the first, second, ... excursions away from x of the random walk
is*}.
10. For any t > 0 let a(t) = sup{r : r < t,W{r) = 0} and f3(t) = inf{r :
t > t,W(r) = 0}. Then the path {Wt(s);a{t) < s < (l(t)} is called an
excursion of W(-).
11. fn is the number of those terms of Si, 52,..., Sn which are positive or which
are equal to 0 but the preceding term of which is positive.
12. 0(n) = #{k : 1 < k < n, Sk-iSk+i < 0} is the number of crossings.
13. R(n) = max{A; : k > 1 for which there exists a0<j<n-k such that
f@).7 + k) = f@, j)} is the length of the longest zero-free interval.
14. r(t) = sup{s : 5 > 0 for which there exists a0< u < t- 5 such that
15. \?(n) = max{A; : 0 < k < n, Sk = 0} is the location of the last zero up to n.
16. i>(t) = sup{5 : 0 < 5 < t, W{s) = 0}.
NON-ACTIVATEDVERSIONwww.avs4you.com
NOTATIONS AND ABBREVIATIONS 7
17. R(n) = max{A: : k > 1 for which there exists a.0<j<n-k such that
M+(j + k) = M+(j)} is the length of the longest flat interval of M? up to
n.
18. f[i) = sup{s : 5 > 0 for which there exists aO< u < t- 5 such that
m+(u + s) = m+(u)}.
19. R*(n) = max{A; : k > 1 for which there exists a 0 < j < n - k such that
20. r*(t) = sup{s : 5 > 0 for which there exists a0< u<f-s such that
m(u + s) = m(u)}.
21. (i(n) is the location of the maximum of the absolute value of a random
walk {Sk} up to n, i.e. fj.(n) is defined by S(fx(n)) = M(n) and fx(n) < n.
If there are more integers satisfying the above conditions then the smallest
one will be considered as /x(n).
22. M{t) = inf{s : 0 < 5 < t for which W(s) = m(t)}.
23. fi+{n) = [nf{k : 0 < k < n for which S{k) = M+{n)}.
24. M+{t) = inf{5 : 0 < 5 < t for which W{s) = m+{t)}.
25. x(n) is the number of those places where the maximum of the random walk
So, Si,..., Sn is reached, i.e. x(n) is the largest positive integer for which
there exists a sequence of integers 0 < kx < k2 < ... < kx(n) < n such that
S{kx) = S(k2) = • - • = S(kx{n)) = M+(n).
Abbreviations
1. r.v. = random variable,
2. i.i.d.r.v.'s = independent, identically distributed r.v.'s,
3. LIL = law of iterated logarithm,
4. UUC, ULC, LUC, LLC, AD, QAD (cf. Section 5.1),
5. i.o. = infinitely often,
6. a.s. = almost surely.
NON-ACTIVATEDVERSIONwww.avs4you.com
Chapter 1
Introduction of Part I
The problems and results of the theory of simple symmetric random walk in Z1
can be presented using different languages. The physicist will talk about random
walk or Brownian motion on the line. (We use the expression "Brownian motion"
in this book only in a non-well-defined physical sense and we will say that the
simple symmetric random walk or the Wiener process are mathematical models of
the Brownian motion.) The number theorist will talk about dyadic expansions of
the elements of [0,1]. The people interested in orthogonal series like to formulate
the results in the language of Rademacher functions. The gambler will talk about
coin tossing and his gain. And a probabilist will consider independent, identicallydistributed random variables and the partial sums of those.
Mathematically speaking all of these formulations are equivalent. In order
to explain the grammar of these languages in this Introduction we present a few
of our notations and problems using the different languages. However, later on
mostly the "language of the physicist and that of the probabilist" will be used.
1.1 Random walk
Consider a particle making a random walk (Brownian motion) on the real line.
Suppose that the particle starts from x = 0 and moves one unit to the left with
probability 1/2 and one unit to the right with probability 1/2 during one time
unit. In the next step it moves one step to the left or to the right with equalprobabilities independently from its location after the first step. Continuing this
procedure we obtain a random walk that is the simplest mathematical model of
the linear Brownian motion.
Let Sn be the location of the particle after n steps or in time n. This model
NON-ACTIVATEDVERSIONwww.avs4you.com
10 CHAPTER 1
clearly implies that
P{Sn+i = tn+i | Sn = in, 5n_i = in_i,..., Si = »i, So = t0 = 0}
= P{Sn+1 = in+1 | Sn = in} = 1/2 A.1)
where i0 = 05 *i, »2) • • • ¦> *n, *n+i is a sequence of integers with |t'i — »o| = |»2 — »i| =
...= |tn+1 — tn| = 1. It is also natural to ask: how far does the particle go away
(resp. going away to the right or to the left) during the first n steps. It means
that we consider
Mn = max \Sk\ resp. M+ = max Sk or M~ = — min S*.
1.2 Dyadic expansion
Let x be any real number in the interval [0,1] and consider its dyadic expansion
x = 0,1
where e< = e,(x) (t = 1,2,...) is equal to 0 or 1. In fact
et = [2'z] (mod 2).
Observe that
\{x : *,-,(*) = 6ush(x) = S2,.. .,ejn(x) = Sn} = 2~n A.2)
where 1 < j\ < j2 < ... < jn;n = 1,2,...; 6i, 62,..., 6n is an arbitrary sequence
of 0's and +l's and A is the Lebesgue measure. Let So = ^o(^) = 0 and Sn =
Sn(x) = n - 2 ?r=i ei(x) (n = 1,2,...). Then A.2) implies
A{x : Sn+1 = «n+1, Sn = in,..., Sl = »!, So = to} = 2~(n+1) A.3)
where t'o = 0, iu i2,..., in+1 is a sequence of integers with |t'i — t'o| = |z*2 — »i| =
...= |tn+i — tn| = 1. Clearly A.3) is equivalent to A.1). Hence any theorem
proved for a random walk can be translated to a theorem on dyadic expansion.A number theorist is interested in the frequency Nn(x) = Z)"=1e,(x) of the
ones among the first n digits of x 6 [0,1]. Since Nn(x) = |(n — Sn(x)) anytheorem formulated for Sn implies a corresponding theorem for Nn(x).
NON-ACTIVATEDVERSIONwww.avs4you.com
INTRODUCTION OF PART I 11
1.3 Rademacher functions
In the theory of orthogonal series the following sequence of functions is well-
known. Let
r(x)-i l if *e [0,1/2),rilxj-\-i if x e [i/2,i],
, f 1 if x€[0,l/4)U[l/2,3/4),r2lXj-\-l if x€[l/4,l/2)U[3/4,l],
, f 1 if x€ [0,1/8) U[l/4,3/8) U[l/2,5/8) U[3/4,7/8),r3lXj-\-l if x€[l/8,l/4)U[3/8,l/2)U[5/8,3/4)U[7/8,l],...
An equivalent definition, by dyadic expansion, is
rn(x) = l-2en(x).
The functions ri(x),r2(x),... are called Rademacher functions. It is a se-
sequence of orthonormed functions, i.e.
Observe that
\{x : rh(x) = 61,rh(x) = 62,.. .,rjn(x) = 6n} = 2~n A.4)
where 1 < j\ < j2 < ... < jn, n = 1,2,...; Si, 62,..., Sn is an arbitrary sequence
of -fl's and —l's and A is the Lebesgue measure. Putting So = S0(x) = 0 and
Sn = Sn{x) = E,n=i »\-(x)(n = 1,2,...) we obtain A.3).
1.4 Coin tossing
Two gamblers (A and B) are tossing a coin. A wins one dollar if the tossingresults in a head and B wins one dollar if the result is tail. Let Sn be the amount
gained by A (in dollars) after n tossings. (Clearly Sn can be negative and So = 0
by definition.) Then 5^ satisfies A.1) if the game is fair, i.e. the coin is regular.
1.5 The language of the probabilistLet Xi, X2,... be a sequence of i.i.d.r.v.'s with
P{Xt = 1} = P{Xt = -1} = 1/2 (» = 1,2,...),
NON-ACTIVATEDVERSIONwww.avs4you.com
12 CHAPTER I
i.e.
P{Xn = *lt Xh = *„..., XJn = U = 2"" A.5)
where 1 < jx < j2 < ... < jn;n = 1,2,... and 6i,62,...,6n is an arbitrarysequence of -f l's and — l's. Let
n
50 = 0 and Sn = ? *k (n = l,2,...).
Then A.5) implies that {Sn} is a Markov chain, i.e.
, Sn-l = »n-l) • ¦ • ) ^1 = *1> ^0 = *0 = 0} =
= *n+i I Sn = in} = 1/2 A.6)
where t'o = 0, i\, i2,..., tn, tn+1 is a sequence of integers with |z'i — t'o| = |»2 — t'i| =
...= |»n+1 -»n| = 1.
NON-ACTIVATEDVERSIONwww.avs4you.com
Chapter 2
Distributions
2.1 Exact distributions
A trivial combinatorial argument gives
THEOREM 2.1
where k = —n, —n + 1,..., n; n = 1,2,...,
= 2* + l}=Bn+IV2"-1 B.2)
where k = —n — 1,—n,... ,n; n = 1,2, —
B.1) and B.2) together give
P{Sn = A:} =
= n (mod2)'
0 otherwise
where k = —n, —n + 1,..., n; n = 1,2, —
Further, for any n = 1,2,..., t 6 -R1 we
B.3)
ESn = 0, E5^ = n, Eexp(*Sn) = (-^—
1 . B.4)
The following inequality (Bernstein inequality) can also be obtained by ele-
elementary methods:
13
NON-ACTIVATEDVERSIONwww.avs4you.com
14 CHAPTER 2
THEOREM 2.2 (cf. e.g. Renyi, 1970/B, p. 387).
2ne2Sn
n>e\ <2expf-
for any n = 1,2,... and 0 < e < 1/4.
For later reference we present also a slightly more general form of the Bern-
Bernstein inequality.
THEOREM 2.3 (cf. Renyi, 1970/B, p. 387). Let X{,X;,... be a sequence ofi.i.d.r.v.'s with
Then for any 0 < e < pq we have
>e\ <2exp
2pq 11 +
where S; = X\ + X\ + • • • + X*n and q = 1 - p.
THEOREM 2.4 (cf. e.g. Renyi, 1970/A, p. 233).
2pql
n+
= k}= [\n-k B.5)
and for any t ? R1
2n
= Eexp(*M2+J =+J = 2~kt
Jk=O
Proof 1 of B.5). (Renyi, 1970/A, p. 233). Let
F+Mn = max } Xj.
J=2
Then
B.6)
>fc= P{X1 = l,M^=A:-l} + = *+¦!}
NON-ACTIVATEDVERSIONwww.avs4you.com
DISTRIBUTIONS 15
-(Pn,Jk-l +Pn,Jk+l) > 1).
Similarly for A: = 0
,= -1,M+ < 1} = i(p^ + pn>0).Pn+1,0 =
Since p10 = pu = 1/2 we get B.5) from B.7) and B.8) by induction.
Proof 2 of B.5). Clearly
P{M+ > A:} = P{5n > A:} + P{5n < k,M+ > k}( n
B.7)
B.8)
j=n (mod 2)
n—j^ 2
Let
Pi(A:) = min{/ : 5, = *},if
i if
i.e. Si for / > p\(k) is the reflection of Sj in the mirror y = k. (Hence the
method of this proof is called reflection principle.) Then
\n-jj=n (mod 2)
n / n
j=n (mod 2)
= 2"" E (n-j)+2-»j'=n (mod 2) j=n (mod 2)
= 2P{5n > A:} + P{5n = A:} = 2~n EJ=Jk
n
n-j
B.9)
which proves B.5).B.6) can be obtained by a direct calculation.
NON-ACTIVATEDVERSIONwww.avs4you.com
16 CHAPTER 2
THEOREM 2.5 For any integers a<O<b,a<b,a<u<bwe have
pn{a, b, u) = P{a < -M- < M+ <b,Sn = u)
= f) qn{u + 2k{b - a)) - ?) qnBb - u + 2k(b - a)) B.10)Jk=-oo fc=-oo
where
0 otherwise
(j = -n,-n + l,...,n;n = 0,1,...)-
Proof. (Billingsley, 1968, p.78). In case n = 0
... { 1 if v = 0 and a2 + b2 > 0,Po(a,b,u) =
jQ otherwise?
and we obtain B.10) easily. Assume that B.10) holds for n — 1 and for any a, b, w
satisfying the conditions of the Theorem. Now we prove B.10) by induction.
Note that pn@,b, w) = pn(a,0,u) = 0 and the same is true for the righthand side
of B.10) (since the terms cancel because qn{j) = <7n(-./))• Hence we may assume
that a < 0 < b. But in this case a + 1 < 0 and b — 1 > 0. Hence by induction
B.10) holds with parameters n — 1, a + 1, b + 1, u and n — 1, a — 1, b — 1, u. We
obtain B.10) observing that
and
pn(a, 6, j/) = -pn-i(a - 1,6 - 1,i/ - 1) + -Pn-i(a + 1,6 + 1,i/ + 1).
THEOREM 2.6 For any integers a < 0 < b and a<u<u<bwe have
P{a < -M~ < M+ < b, u < Sn < u}
f a) < Sn<v + 2k{b-a)}k=-oo
~ E P{2b-v + 2k{b-a) < Sn<2b-u + 2k{b-a)}, B.11)Jk=-oo
NON-ACTIVATEDVERSIONwww.avs4you.com
DISTRIBUTIONS 17
P{a < -M~ <M+<b}
= E P{a + 2k{b-a) <Sn<b + 2k{b-a)}k=-oo
- E ?{b + 2k{b-a) < Sn < 2b - a + 2k{b - a)} B.12)Jk=-oo
and
P{Mn < 6} = E P(D* " !N <Sn< D*Jfc=-oo
Sn< DA: + 3N}. B.13)Jk=-oo
B.11) is a simple consequence of B.10), B.12) follows from B.11) taking u =
a,v = b and B.13) follows from B.12) taking a = -b.
To evaluate the distribution of Ii(n,a) (i = 1,2,3,4,5) seems to be very
hard (cf. Notations to the Increments). However, we can get some information
about the distribution of Ji(n,a).
LEMMA 2.1 (Erdos - Revesz, 1976).
p{n+j,n) = P{Il{n + j,n) = n) = 3-^T (j = 0,1,2,..., n).
Clearly p(n + j, n) is the probability that a coin tossing sequence of lengthn + j contains a pure-head-run of length n.
Proof. Let
A = {I^n + j,n) = n} and Ak = {Sk+n - Sk = n}.
Then
A = Ao + A0Ai + A0AiA2 + • • • + A0Ai • ¦ • Aj^iAj= Ao + A0Ai + AxAi H h Aj-iAj.
Since P(A0) = 2~n and PiAoAi. ..M>i) = 2~n-1 for any t = 1,2,... ,j- 1 we
have the Lemma.
The next recursion can be obtained in a similar way.
NON-ACTIVATEDVERSIONwww.avs4you.com
18 CHAPTER 2
LEMMA 2.2 For any j = 1,2,... we have
Bn + j, n) = n} = pBn + j, n)
2n+l'
A - p(n + 2, n))^ + • • • + A - P(n + j- 1, n)) —
In case j < n we obtain
In some cases it is worthwhile to have a less exact but simpler formula. For
example, we have
LEMMA 2.3 (Deheuvels - Erdos - Grill - Revest, 1987).
{j + 2J-*-J - {j + 2J2�n~2 < P{h{n + j,n) = n) < {j + 2J"n� B.14)
for anyn = 1,2,... ;j = 1,2,....
The idea of the proof is the same as those of the above two lemmas. The
details are omitted.
The exact distribution of Zn (cf. Notations to the Increments) is also known,
namely:
THEOREM 2.7 (Szekely - Tusnady, 1979).
where
Remark 1. Csaki, Foldes and Komlos A987) worked out a very general method
to obtain inequalities like B.14) .Their method gives a somewhat weaker result
than B.14). However, their result is also strong enough to produce most of the
strong theorems given later (cf. Section 7.3).
NON-ACTIVATEDVERSIONwww.avs4you.com
DISTRIBUTIONS 19
2.2 Limit distributions
Utilizing the Stirling Formula
(where 0 < an < 1) and the results of Section 2.1, the following limit theorems
can be obtained.
THEOREM 2.8 (e.g. Renyi, 1970/A, p. 208). Assume that for some 0 < e <
1/2 the inequality en < k < A — e)n is satisfied. Then
(n\ . _2"«jr>
where K = k/n and d(K) = Klog2K + A - K) log2A - K). If we also assume
that \k-n/2\= o{n2lz) then
Especially
2nJ-2nThe next theorem is the so-called Central Limit Theorem.
THEOREM 2.9 (Gnedenko - Kolmogorov, 1954, §40).
<x}- *(z)| < 2m�/2.
A stronger version of Theorem 2.9 is the so-called Large Deviation Theorem:
THEOREM 2.10 (e.g. Feller, 1966, p. 517).
Pjn-1/2^ < -xn} F{n-l'*Sn > xn}hm —i—— = hm —*-^y—.—-n—oo $(-Xn) n—oo 1 — $(xn)
provided that 0 < xn = o(n1/6).
Theorem 2.10 can be generalized as follows:
NON-ACTIVATEDVERSIONwww.avs4you.com
20 CHAPTER 2
THEOREM 2.11 (e.g. Feller, 1966, p. 517). Let X{,X;,... be a sequence
of i.i.d.r.v. 's with
EX' = 0, E(X'J = I, Eexp(*X;) < oo
for all t in some interval \ t \< to. Then
P{n-^S^ < -xn} F{n-V*S; > xn}lim —
r = lim ——r = 1
$() 1 $()n-oo
provided that 0 < xn = o{n1/6) where S^ = X{ + X\ + ¦ • • + X*n.
THEOREM 2.12 (e.g. Renyi, 1970/A, p. 234).
lim F{n-1'2M+ < x} = P{\N\ < x} = 2S(z) - 1n—»oo
uniformly in x G R1 where N 6 iV@, l). Further,
lim E(n�/2Mn+) = B/ttI/2.
THEOREM 2.13
lim P{n-^2Mn <x} = G{x) = H{x)n—'oo
uniformly in x ? R1. Further,
v P{n/Mn > xn} Fjn^Mn < x}lim 7-7—r
= lim ——rr — 1n-00 1 - G(xn) n-°° H{1)
where
provided that 0 < xn = o{nxl*). Consequently for any e > 0
?{n-l'2Mn > xn} > A - e){l - G{xn))
>2A-
NON-ACTIVATEDVERSIONwww.avs4you.com
DISTRIBUTIONS 21
> xn} < A + e)(l - G(xn))
4A -e) f / 7T2 2\ 1 / 9tt2 2\1 . .
> A—I ^exp ^-T,iJ-
rxp (—i-^JJ B-18)
if 0 < xn = ©(n1/6) and n is 6«*flr enough.
Remark 1. As we claimed G{x) = #(z) however in Theorem 2.13 the asymp-
asymptotic distribution in the form of G(-) is proposed to be used when x is big. When
x is small H(-) is more adequate.Finally we present the limit distribution of Zn.
THEOREM 2.14 (Foldes, 1975, Goncharov, 1944). For any positive integer k
we have
P{Zn - [lg N}<k}= exp(-2-(fc+1)-^Ar>) + o(l)
where {lg iV} = lg iV - [lg N].
Remark 2. As we have mentioned earlier the above Theorems can be provedusing the analogous exact theorems and the Stirling formula. Indeed this method
(at least theoretically) is always applicable, it often requires very hard work.
Hence sometimes it is more convenient to use characteristic functions or other
analytic methods.
NON-ACTIVATEDVERSIONwww.avs4you.com
Chapter 3
Recurrence and the Zero-One Law
3.1 Recurrence
One of the most classical strong theorems on random walk claims that the particlereturns to the origin infinitely often with probability 1. That is
RECURRENCE THEOREM (Polya, 1921).
P(Sn = 0 i.o.) = 1.
We present three proofs of this theorem. The first one is based on the followinglemma:
LEMMA 3.1 Let 0 < i < k. Then for any m>i we have
p@, t, k) = P{min{j : j > m, 5, = 0} < min{j : j > m, Sj = k} \ Sm = i}= k~1(k-i), C.1)
i.e. the probability that a particle starting from i hits 0 before k is k-1(k — i).
Proof. Clearly we have
p@,0,*) = l, p@,M)=0.
When the particle is located in t then it hits 0 before A: if
(i) either it goes to i — 1 (with probability 1/2) and from i — 1 goes to 0 before
A: (with probability p@,t - l,k)),
(ii) or it goes to i + 1 (with probability 1/2) and from » + 1 goes to 0 before A;
(with probability p@,t + l,k)).
23
NON-ACTIVATEDVERSIONwww.avs4you.com
24 CHAPTER 3
That is
p@, i, Ar) = ip(O,» - 1,*) + ip(O,» + 1,*)
(i = 1,2,..., A: — 1). Hence p@, i, k) is a linear function of i, being 1 in 0 and 0
in k, which implies C.1).
Proof 1 of the Recurrence Theorem. Assume that S\ = 1, say. By C.1)for any e > 0 there exists a positive integer no = no(e) such that p@, l,n) =
1 - 1/n > 1 - ?¦ if n > n0- Consequently the probability that the particle returns
to 0 is larger than 1 — e for any e > 0. Hence the particle returns to 0 with
probability 1 at least once. Having one return, the probability of a second return
is again 1. In turn it implies that the particle returns to 0 infinitely often with
probability 1.
Proof 2 of the Recurrence Theorem. Introduce the following notations
Po = 1,
P» = *{SU = 0} = 2~U^) (* = 1,2,...),
A2k = {S2k = 0, S2Jk-2 # 0,52Jk_4 # 0,..., S2 ? 0},q2k =
Jk=O
oo
(Note that q2k is the probability of the event that the first return of the particleto the origin occurs in the BA:)-th step but not before.)
Since p2k « (ttA;)�/2 (cf. Theorem 2.1) we have
limP(z) = oo.
Observe thatJk—1
{S2k = 0} = A2k + Y, A2k-2j{S2k = 0}
and
Y{A2k-2j{S2k = 0}} = q2k-2jp2j.
NON-ACTIVATEDVERSIONwww.avs4you.com
RECURRENCE AND THE ZERO-ONE LAW 25
Hence
@) Po = 1,
A) p2 = 92,
.(ii) P4 = 94 + 92P2,
(iii) p6 = 96 + 94P2 + 92P4,• • •
(k) P2Jk = 92Jfc + 92Jk-2P2 H H 92P2Jk-2,' ' '
Multiplying the Jfc-th equation by z2fc(|2| < 1) and summing up to infinity we
obtain
P(z)=P(z)Q(z)+l,i.e.
Q{z) = 1 " ^T and Jim Q(z) = 1.
Since Q(l) = YtkLitek = 1 is the probability that the particle returns to the
origin at least once we obtain the theorem.
3.2 The Zero-One Law
The above two proofs of the Recurrence Theorem are based on the fact that if
P{5jk = 0 at least for one n} = 1
then
P{5n = 0 i.o.} = 1.
Similarly one can see that if P{5n = 0 at least for one n) were less than 1 then
P{5n = 0 i.o.} would be equal to 0. Hence without any calculation one can
see that P{5n = 0 i.o.} is equal to 0 or 1. Consequently in order to prove the
Recurrence Theorem it is enough to prove that P{5n = 0 i.o.} > 0.
In the study of the behaviour of the infinite sequences of independent r.v.'s
we frequently realize that the probabilities of certain events can be only 0 or 1.
Roughly speaking we have : let Y\,Y2,... be a sequence of independent r.v.'s.
Then, if A is an event depending on Yn,Yn+\,... (but it is independent from
Y\, Yi,..., Yn-i) for every n, it follows that the probability of A equals either 0
or 1. More formally speaking we have
ZERO-ONE LAW (Kolmogorov, 1933). Let YuY2i... be independent r.v.'s.
Then if A ? Q is a set measurable on the sample space of Yn, Fn+i,... for every
n, it follows that
P{A) =0 or P(A) = 1.
NON-ACTIVATEDVERSIONwww.avs4you.com
26 CHAPTER 3
Example 1. Let YUY2,... be independent r.v.'s. Then ?,°li Y" converges a.s.
or diverges a.s.
Having the Zero-One Law we present a third proof of the Recurrence Theo-
Theorem. It is based on the following:
LEMMA 3.2 For any —oo < a < b < +oo we have
P{liminf Sn = a} = PlimsupSn = b} = 0.n—oo n-»oo
Proof is trivial.
Proof 3 of the Recurrence Theorem. Lemma 3.2, the Zero-One Law and
the fact that Sn is symmetrically distributed clearly imply that
P{liminf Sn = -oo} = P{limsup5n = oo} = 1 C.2)n—»oo
n—»oo
which in turn implies the Recurrence Theorem.
Note that C.2) is equivalent to the Recurrence Theorem. In fact the Recur-
Recurrence Theorem implies C.2) without having used the Zero-One Law.NON-ACTIVATEDVERSIONwww.avs4you.com
Chapter 4
From the Strong Law of Large Numbers
to the Law of Iterated Logarithm
4.1 Borel - Cantelli lemma and
Markov inequality
The proofs of almost all strong theorems are based on different forms of the Borel
- Cantelli lemma and those of the Markov inequality. Here we present the most
important versions.
BOREL - CANTELLI LEMMA 1 Let AUA2,... be a sequence of events forwhich Y%Li P(Ai) < oo. Then
P{limsup An} = P ( fl f>4 = P(^n i.o.) = 0,n-°° ln=l.=n )
i.e. with probability 1 only a finite number of the events An occur simultaneously.
BOREL - CANTELLI LEMMA 2 Let AUA2,... be a sequence of pairwiseindependent events for which Y^=\ P(-^n) = oo. Then
P{limsup An) = 1,n—»oo
i.e. with probability 1 an infinite number of the events An occur simultaneously.
BOREL - CANTELLI LEMMA 2*(Spitzer, 1964). Let AUA2,... be a se-
27
NON-ACTIVATEDVERSIONwww.avs4you.com
28 CHAPTER 4
quence of events for which
T P(/U = oo and liminf *=^=* -j-< C (C > 1).
Then
P{limsupAn} >C~l.n—»oo
MARKOV INEQUALITY Let X be a non-negative r.v. with EX < oo. Then
for any A > 0
P{X > XEX} < \.As a simple consequence of the Markov inequality we obtain
CHEBYSHEV INEQUALITY Let X be an r.v. with EX2 < oo. Then forany A > 0
P{|X - EX| > A(E(X - EXJI/2} = P{(X - EXJ > A2E(X - EXJ} < —.
Similarly we get
THEOREM 4.1 Let X be an r.v. with E(exp(*X)) < oo for some t > 0. Then
for any A > 0 we have
P{X > A} = P{exp(*X) > eAt} < 5^-.Borel - Cantelli lemmas 1 and 2 and Markov inequality can be found practi-
practically in any probability book (see e.g. Renyi, 1970/B).
4.2 The strong law of large numbers
THEOREM OF BOREL A909).
lim n^Sn = 0 a.s. D.1)
Remark 1. Applying this theorem for dyadic expansion we obtain
limn—oo n~1Nn(x) = 1/2 for almost all x ? [0,1]. In fact the original theorem
NON-ACTIVATEDVERSIONwww.avs4you.com
STRONG LAW AND LIL 29
of Borel was formulated in this form. Borel also observed that if instead of the
dyadic expansion we consider t-s.dk. expansion (t = 2,3,...) of x G [0,1] and
Nn(x,s,t) (s = 0,1,2,...,t - l,t = 2,3,...) is the number of s's among the
first n digits of the 4-adic expansion of x, then
Nn{x,s,t)lim
n—»oo
= 0 (s = 0,l,2,...,*-l;* = 2,3,...) D.2)n t
for almost all x. Hence Borel introduced the following:
Definition. A number x G [0, l] is normal if for any s = 0,1,2,..., t — 1; t =
2,3,... D.2) holds.
The above result easily implies
THEOREM 4.2 (Borel, 1909).
Almost all x G [0,1] are normal.
It is interesting to note that in spite of the fact that almost all x G [0,1] are
normal it is hard to find any concrete normal number.
Proof 1 of D.1). (Gap method). Clearly (cf. B.4))
En-'S,, = 0, En�S2 = n�.
Hence by Chebyshev inequality for any e > 0
Pdn^Snl >e}< n~xe-2
and by Borel - Cantelli lemma 1
n~2Sn3 —*¦ 0 a.s. (n —> oo).
Now we have to estimate the value of Sk for the fc's lying in the gap, i.e. between
n2 and (n + lJ. If n2 < A: < (n + IJ then
lAT^I = \n-*Sn*n*k-x + A:�E, - Sn,)| < \n~2Sn,\ + k~l{{n + lJ - n2).Since both members of the right hand side tend to 0, the proof is complete.
Proof 2 of D.1). (Method of high moments). A simple calculation gives
and again the Markov inequality and the Borel - Cantelli lemma imply the the-
theorem.
As we will see later on most of the proofs of the strong theorems are basedon a joint application of the above two methods.
NON-ACTIVATEDVERSIONwww.avs4you.com
30 CHAPTER 4
4.3 Between the Strong Law of Large Numbers
and the Law of Iterated Logarithm
The Theorem of Borel claims that the distance of the particle from the originafter n steps is \Sn\ = o(n) a.s. It is natural to ask whether a better rate can be
obtained. In fact we have
THEOREM OF HAUSDORFF A913). For any e > 0
lim n-1/2-eSn = 0 a.s.n—»oo
Proof. Let K be a positive integer. Then a simple calculation gives
ES2* = 0{nK).
Hence the Markov inequality implies
> nK+eK} < 0{n~eK).
If K is so big that eK > 1 then by the Borel - Cantelli lemma we obtain the
theorem. (The method of high moments was applied.)Similarly one can prove
THEOREM 4.3
lim sup .
n
<1 a.s.
n—oo n1/2logn
Proof. By B.4) we have
Eexptn-^S^e1/2 (n^oo).
Hence
= P{exp(n-1/25n) > exp((l + e) logn) = n1+e} < n��/
if n is big enough. Consequently
lim sup .
w<1 a.s.
n—oo n^'logn
and the statement of the Theorem follows from the symmetry of Sn.The best possible rate was obtained by Khinchine. His result is the so-called
Law of Iterated Logarithm (LIL).
NON-ACTIVATEDVERSIONwww.avs4you.com
STRONG LAW AND LIL 31
4.4 The LIL of Khinchine
LIL OF KHINCHINE A923).
Iimsup6n5n = Iimsup6n|5n| = limsup6nMnn—»oo n—»oo n—»oo
= \imsup bnM+ = limsup6nM~ = 1 a.s.
n—»oo n—»oo
where bn = Bnloglogn)~1/2.
Proof. The proof will be presented in two steps. The first one gives an upper
bound of lim supn_oo bnMn, the second one gives a lower bound of lim supn_oo bnSn.These two results combined imply the Theorem.
Step 1. We prove that for any e > 0
limsup6nMn < 1 + ?¦ a.s.
n—oo
By B.16) we obtain
P{Mn > (l + e)^1} <exp(-(l + ?:)loglogn) = (logn)-1� D.3)
if n is big enough. Let nk = [0*] @ > 1). Then by the Borel - Cantelli lemma
we get
Mnk < A + e)b~l a.s.
for all but finitely many k. Let n^ <n < njk+1. Then
Mn < Mnt+l < A + e)b~lk+l < A + 2s)b-lk < A + 2e)b~l a.s.nt+l
< A + e)bk+l <
provided that 0 is close enough to 1.
We obtain
limsup6nTn < 1n—»oo
where Tn is any of Sn, | Sn |, Afn, Af+, Af".
Observe that in this proof the gap method was used. However, to obtain
inequality D.3) it is not enough to evaluate the moments or the moment gener-
generating function of Mn (or that of Sn) but we have to use the stronger result of
Theorem 2.13.
Step 2. Let nk = [0*] @ > 1). Then for any e > 0
" Snk) >l-s}>
NON-ACTIVATEDVERSIONwww.avs4you.com
32 CHAPTER 4
if A: is big enough. Since the events {bnk+1 {Snk+1-Snt) > A -e)} are independent,we have by Borel - Cantelli lemma 2
">• a.8.
Consequently
Applying the result proved in Step 1 we obtain
if A; and 0 are big enough. Hence limsupn_oo bnSn > 1 — e a.s. for any e > 0,which implies the Theorem.
Note that the above Theorem clearly implies
limsupSn = oo, liminfSn = —oo a.s.,n—oo
n~*°°
which in turn implies the Recurrence Theorem of Section 3.1.
For later references we mention the following strong generalization of the LIL
of Khinchine.
LIL OF HARTMAN - WINTNER A941). Let YUY2,... be a sequence ofi.i.d.r.v.'s with
= 0, E^2 = 1.
Then
limsup&n(Yi +Y2 + -" + Yn) = 1 a.s.
n—»oo
Remark 1. Strassen A966) also investigated the case EYi2 = oo. In fact he
proved that if Yi, Y2,... is a sequence of i.i.d.r.v.'s with EYi = 0 and EYj2 = oo
then
Iimsup6n|yi + Y2 H h Yn\ = oo a.s.
n—»oo
Later Berkes A972) has shown that this result of Strassen is the strongest possibleone in the following sense: for any function f(n) with limn_oo f(n) = 0 there
exists a sequence Y\, Y2,... of i.i.d.r.v.'s for which EYi = 0, EYt2 = oo and
lim 6n/(n)|yi =0 a.s.
NON-ACTIVATEDVERSIONwww.avs4you.com
Chapter 5
Levy Classes
5.1 Definitions
The LIL of Khinchine tells us exactly (in certain sense) how far the particlecan be from the origin after n steps. A trivial reformulation of the LIL is the
following:(i) for any s > 0
Sn < A + ^b'1 a.s. for all but finitely many n
and
(")Sn > A - ?)Kl i-o- a.s.
Having in mind this form of the LIL, Levy asked how the class of those
functions (or monotone increasing functions) /(n) can be characterized for which
Sn < f(n) a.s.
for all but finitely many n. (i) tells us that A + e)b~l is such a function for
any e > 0 and (ii) claims that A — e)b~l is not such a function. The LIL does
not answer the question whether b'1 is such a function or not. However, one
can prove that b'1 is not such a function but Bn(log log n + 3/2 log log log n)I/2belongs to the mentioned class. In order to formulate the answer of Levy'squestion introduce the following definitions.
Let {Y(t),t > 0} be a stochastic process then
Definition 1. The function ax(t) belongs to the upper-upper class of {Y(t)} (at €
UUC(y(*))) if for almost all w € ft there exists aio = to{u) > 0 such that
Y(t) <ai(<) if t> t0.
33
NON-ACTIVATEDVERSIONwww.avs4you.com
34 CHAPTER 5
Definition 2. The function a2{t) belongs to the upper-lower class of
(a2 G ULC(F(i))) if for almost all w G 0 there exists a sequence of positivenumbers 0 < tv = ti(u) < t2 = t2{uj) < ... with tn —*• oo such that Y(ti) >
Definition 3. The function a3(t) belongs to the lower-upper class of
(a3 G LXJC(Y(t))) if for almost all u G Cl there exists a sequence of positivenumbers 0 < tx = ti[u) < t^ = ^(w) < ••• with tn —+ oo such that
Definition 4. The function aA{t) belongs to the lower-lower class of
(a4 G LLC(F(?))) if for almost all u G n there exists a t0 = to(uj) > 0 such that
Y{t) > aA{t) \it > t0.
Let Yx, F2,... be a sequence of random variables then the Levy classes UUC(Yn),ULC(rn), LUC(rn), LLC(Fn) of {Yn} can be defined in the same way as it was
done above for Y(t).We introduce two further definitions strongly connected with the above four
definitions of the Levy classes.
Definition 5. The process Y(t) is asymptotically deterministic (AD) if there
exists a function ax{t) G UUC(F(*)) and a function a4(t) G LLC(Y(t)) such that
limt-,00 |a4@ -ai(*)| = 0.
Consequently
lim |a4m - Y{t)\ = lim \ai{t) - Y(t)\ = 0 a.s.t—»oo t—*oo
Definition 6. The process Y(t) is quasi AD (QAD) if there exists a function
ax(<) G U\JC(Y(t)) andafunction aA{t) G LLC(F(*)) such that limsup^^ \aA{t)-ai{t)\ < oo.
The definition of AD resp. QAD sequences of r.v.'s can be obtained by a
trivial reformulation of Definitions 5 and 6.
Remark 1. Clearly the UUC(rn) resp. the XJXJC(Y(t)) is the complementerof the ULC(Fn) resp. the ULC{Y{t)) and similarly the LUC(rn) resp. the
LUC(y(*)) is the complementer of the LLC(yn) resp. the LLC(F(*)).
NON-ACTIVATEDVERSIONwww.avs4you.com
LEVY CLASSES 35
5.2 EFKP LIL
Now we formulate the celebrated Erdos A942), Feller A943, 1946), Kolmogorov- Petrowsky A930 - 35) theorem.
EFKP LIL The nondecreasing function a(n) € UUC(rn) if and only if
~a{n) ( a2{n)\^J < oo
n=l"
where Yn is any of n�/2S^n'1'2 \ Sn |,
This theorem completely characterises the UUC(Fn) if we take into consider-
consideration only nondecreasing functions and it implies
Consequence 1. For any e > 0
Sn < (nBloglogn + C + e) logloglogn)I/2 a.s. E.1)
for all but finitely many n. Further
Sn > (nBloglogn + 3logloglogn)I/2 i.o. a.s. E.2)
Here we present the proof of Consequence 1 only instead of the proof of EFKP
LIL (cf. Remark 1 at the end of Section 5.3).
Proof of Consequence 1. The proof will be presented in two steps.
Step 1. We prove that for any e > 0 and for all but finitely many n
Mn < (nB log log n + C + e) log log log n)I/2 a.s., E.3)
which clearly implies E.1). By B.16) we obtain
P{Mn > (nB log2 n + C + e) log3 n)I/2}
< 4A+4-1-! L_B_=3i+fLi-n *.. E.4)
V 2tt J2 log2 n lo8n (log2 n) a V ^ loSn (Iog2 n) 2
Let
f / * Mnjk = exp
Then by the Borel - Cantelli lemma we get
Mnk < {nkB log2 nk + C + e) log3 n^)I/2 a.s.
NON-ACTIVATEDVERSIONwww.avs4you.com
36 CHAPTER 5
for all but finitely many k. Let nk < n < nk+l. Then
Mn < Mnk+l < (nJk+1Blog2 nk+l + C + e) log3 n^)I'2< (nJkB log2 nk + C + 2e) log3 n*)I72 E.5)
which implies E.3).
Step 2. Introduce the following notations
- = [expy?(n) = Blog2n + 3log3nI/2,
<p*{n) = Blog2n + 6log3nI/2,An = {n
Then clearly
T(Ank) = Oik-1 (log k)-1),l(log k)-*'2),
and for any j < k — j + m we have
V(AnjAnk) = P {*>>,) > nJ1/J5Bj. > ^(n,),^172^ > <p(nk)}+ T{nJ1/2Snj > ^(n^nl^S^ > <p(nk)}
<V{tp*{ni)>nJ1/iSn.>ip{ni)
P
_[5ni- Sn. I n^XP {
'> J <p(nk) -
+ o (r^iogj)-*'*) = [o (rx(\ogjyy) - o
+ o(y-1(iogy)-5/2)where
and t =
-
n.
NON-ACTIVATEDVERSIONwww.avs4you.com
LEVY CLASSES
Observe that
and
Hence
Since
and
with
|3 log log j4
> <p{nk)
4 logj
'y/nk-Jnj y/*j 3loglogy"\y/nk
-
rij y/nk-
rij 4 log;
> i^3! if 1< x < 4fx-l 3
<x-\
x = — = exp
if x > 1
j \
= exp
for any j big enough, we obtain
m
m. log 1 + -
j V j
log; log(y + m
37
(-
rij-
3 \rij
1> -
~
3
m1/2
m. log 1 + —
j V j
iog(y + m) logy iog(y
1/21 ( m
4 \iogy
if 1 < m < (log 4) logj, and
1 m
1 ,> 1 —
exp—
m)
if m > (log4) logj.
NON-ACTIVATEDVERSIONwww.avs4you.com
38 CHAPTER 5
Similarly
1/2
Hence
? > O(m1/2) if 2 log log; < m < (log 4) log; E.6)
and4
^()if (l4) li < m < (i°gi) log log y.
In case m > log j(log log./) we obtain
and;+log;(loglog;)
? P(An>AnJ < O (j-'ilogj)-1 log log j) . E.8)
Having E.7) and E.8) a simple calculation gives
P(AnyAnJ=o((loglogiVJ)
arid
EP(il»J=O(loglogJV).
Hence the Borel - Cantelli lemma 2* of Section 4.1 and the Zero-One Law of
Section 3.2 imply the theorem. A simple consequence of EFKP LIL is
THEOREM 5.1 The nonincreasing function -c(n) € LLC(n-1/2S'n) if and
only if Ix{c) < oo.
The Recurrence Theorem of Section 3.1 characterizes the monotone elements
of LLC(|5n|). In fact
THEOREM 5.2 A monotone function d(n) € LLC(|5n|) if and only if d(n) < 0
for any n big enough.
Remark 1. For the role of Kolmogorov in the proof of EFKP LIL see BinghamA989).
NON-ACTIVATEDVERSIONwww.avs4you.com
LEVY CLASSES 39
5.3 The laws of Chung and Hirsch
The characterization of the lower classes of Mn and M+ is not trivial at all. We
present the following two results.
THEOREM OF CHUNG A948). The nonincreasing function a(n) e
LLC(n-^M,,) if and only if
h{a) = f; »-'(a(n))-'exp (-?«"») < <*>•
THEOREM OF HIRSCH A965). The nonincreasing function 0(n) €
LLC(n�/2M+) if and only if
Note that Theorem of Chung trivially implies
«-»«> \ n J V8
E.9) is called the "Other LIL".
Remark 1. The proof of EFKP LIL is essentially the same as that of Con-
Consequence 1. However, it requires a lemma saying that if a monotone function
/(•) E UUC(Sn) then f{n) > 6;1 /2 and if /(•) € ULC(S'n) then f(n) < 2b~x.
The proofs of Theorems of Chung and Hirsch are also very similar to the above
presented proof (cf. Consequence 1 of Section 5.2). However, instead of B.15)and B.16) one should apply B.17) and B.18).
5.4 When will Sn be very big?
We say that Sn is very big if Sn > b~l. EFKP LIL of Section 5.2 says that Sn is
very big i.o. a.s. Define
a{n) = max{k : 0 < k < n, Sk > &*1}, E.10)
i.e. <x(n) is the last point before n where Sk is very big. The EFKP LIL also
implies that a(n) = n i.o. a.s. Here we ask: how small can a(n) be? This
question was studied by Erdos - Revesz A989). The result is:
NON-ACTIVATEDVERSIONwww.avs4you.com
40 CHAPTER 5
THEOREM 5.3
(log log n)'/2 a(n)log =-G a.s.
n"-*00 (log log log n) log n n
where C is a positive constant with
ry-2 < fi < ol4
Equivalentlya(n) > nx~Sn a.s.
for all but finitely many n where
log log log n
n=(log log nI/*-
The exact value of C is unknown.
Clearly one could say that Sn is very big if
(i) Sn> (l-e)BnloglognI/2 @ < e < 1) or
(ii) Sn > Bn(log log n + | log log log n)) , e.t.c.
These definitions of "very big" are producing different a's instead of the one
defined by E.10). It is natural to ask: what can be said about these new a's?
It is also interesting to investigate the time needed to arrive from a very bigvalue to a very small one. Introduce the following notations: let
a.\ = min{A;: k > 3, Sk > b^1},01 = min{A;: k > a^, S^ < —b^1},oli = min{fc '• k > 0i, Sk > bfr },
02 — min{A;: k > a-i, Sk < — b^1},...
Define a sequence of integers {nk} by
_ c », r». *+! l iu o q ^n\ — o, n/c — itxjfc—i-x j \"'
—
"> **> • • •)•
Then by Theorem 5.3 between n^ and nk+i there exist integers j and / such that
Sj > bj1 and Si < -6,�.
Hence we have
NON-ACTIVATEDVERSIONwww.avs4you.com
LEVY CLASSES 41
THEOREM 5.4
0k < nk a.s.
for all but finitely many k.
Very likely a lower estimate of 0k is also close to nk but it is not proved.
Remark 1. The limsup of the relative frequency of those i's A < i < n) for
which Si > A — eNfrl IS investigated in Section 8.1.
5.5 A theorem of Csaki
The Theorem of Chung (Section 5.3) implies that with probability 1 there are
only finitely many n for which
or in other words there are only finitely many n for which simultaneously
/ 2 \ i/2 / 2 \ V2
(-^M and8loglogn; ^y
for any 0 < e < 1. At the same time Theorem of Hirsch (Section 5.3) impliesthat with probability 1 there are infinitely many n for which
B\ !/2
( 8 log log n
In fact there are infinitely many n for which
AC < ?
Roughly speaking this means that if M+ is small (smaller than A—?){8i*\" n)then M~ is not very small (it is bigger than A - e)(81o^gwI/2 ) provided that
n is big enough. Csaki A978) investigated the question of how big M~ must be
if M+ is very small. His result is
THEOREM 5.5 Let a{n) > 0,6(n) > 0 be nonincreasing functions. Then
<a(n)n^ and M~ <b{n)n^ i.o.} = \\ ^ A(a(n),6(n)) = oo,n ~ v ' J
[ 0 otherwise
NON-ACTIVATEDVERSIONwww.avs4you.com
42 CHAPTER 5
where
and c(n) = a(n) + b[n).
The special case a(n) = b(n) of this theorem also gives Chung's theorem. For-
Formally this theorem does not contain Hirsch's theorem.
In order to illustrate what Csaki's theorem is all about we present here two
examples.
Example 1. Put
a{n) =C(loglogn)-1'2 @ < C < ir/y/s)
and
b{n) = ?>(loglogn)-1/2 (D > 0).
Then
J4(a(n), 6(n)) < oo if D < tt/v^ - C
and
/4(a(n), 6(n)) =oo if D > Ttjyfc - C.
Applying Csaki's theorem this fact implies that the events
(\1/2 / \ 1/2
—=—) and M-<D[-^—)loglogn/
n
Vloglogn/
occur infinitely often with probability 1 if D > n/v2 — C. However, it is not so
if D < ir/y/2 — C. That is to say if n is big enough and
/ \i/J
: <c[ —— (o < c < ifiVs),\loglogn/
then it follows that
'n —
/ \ 1/2
> D I for any 0 < D < tt/v^ - C.
\\og\ognj
Example 2. Put
a(n) = (logn)-a @ < a < 1)and
b{n) = J?;(loglogn)-1/2 {E > 0).
NON-ACTIVATEDVERSIONwww.avs4you.com
LEVY CLASSES 43
Then
J4(a(n),6(n)) < oo if 0<a<l and E < ttBA - a))~1/2and
J4(a(n),6(n)) =00 if 0< a < 1 and E > ttBA - a))�/2.Observe also that
^((lognJ-^FOogloglogn)-1/2) < oo if F<ir/V2and
J^OognJ^FOogloglogn)-1/2) = oo if F
Applying Csaki's theorem this fact implies that the events
{M+ <n1/2(logn)-a and M~ <
resp.
{M+ < n^logn)-1 and M" < Fn^logloglogn)�/2}occur infinitely often with probability one if E > 7rB(l — a))�/2 resp. F > n/y/2.However, it is not so if E < 7rB(l - a))�/2 resp. F < n/y/2.
As we have mentioned already, Csaki's theorem states that if one of the
r.v.'s M? and M~ is very small than the other one cannot be very small. It is
interesting to ask what happens if one of the r.v.'s M+ and M~ is very big. In
Section 8.1 we are going to prove Strassen's Theorem 1, which easily implies that
for any e > 0 the events
{Mn+>i^BnloglognI/2 and M" > ^BnloglognI/2}occur infinitely often with probability 1, but of the events
{Mn+>^BnloglognI/2 and M" > ^BnloglognI/2}only finitely many occur with probability 1. In general one can say
THEOREM 5.6 For any e > 0 and l/3<q<l the events
: > A - e)gb-1 and M~ > A - <0^Voccur infinitely often with probability 1, but of the events
x and M" > A+ e)ionly finitely many occur with probability 1.
NON-ACTIVATEDVERSIONwww.avs4you.com
44 CHAPTER 5
As a trivial consequence of Theorems 5.5 and 5.6 we obtain
THEOREM 5.7 Consider the range M*n = M++M~ of the random walk{Sn}.Then for any e > 0 we have
(l + e)BnloglognI/2 € UUC(M^),
A - e)BnloglognI/2 € ULC(M^),'2ir n
8 log log n)
v '
\ 8 log log n)
Theorems 5.5 and 5.6 describe the joint behaviour of M+ and M~. We also
ask what can be said about the joint behaviour of Sn and M~ (say). In order to
formulate the answer of this question we introduce the following notations.
Let 7{n),6{n) be sequences of positive numbers satisfying the following con-
conditions:
7(n) monotone,
6{n) | 0,
) T oo,
nxl26{n) T oo.
Further let f(n) = nl/2tjj{n) ? ULC(S'n) with t/>(n) | oo. Define the infinite
random set of integers
Then we have
THEOREM 5.8 (Csaki - Grill, 1988). For any f(n) = nx^{n) € ULC(Sn)the function
g{n) = nx'h{n) € UUC(M",n € f)
f{n)+2g{n)e\JXJC{Sn).
nx/2S{n) €LLC(Mn-,n€<r)
w-pf-^<-.
if and only
Further,
if and only
if
ifOO
En=l
n1/2(
*(»)n
NON-ACTIVATEDVERSIONwww.avs4you.com
LEVY CLASSES 45
Remark 1. nl/2-y(n) e UUC(M~,n e c) means that nl/2-y(n) > M~ a.s. for
all but finitely many such n for which n € <;. In other words the inequalitiesSn > f{n) and M~ > n1/'27(n) simulteneously hold with probability 1 only for
finitely many n.
Consequence 1. Let V(n) = min(M+,M"). Then /(n) € UUC(Vr) if and only
if3/(n) €UUC(S'n).
Remark 2. Theorem 5.6 in case q = 1/3 follows from the above Consequence1. For other g's A/3 < q < 1) Theorem 5.8 implies Theorem 5.6.
Example 3. Let f(n) = (B - ^nloglognI/2 @ < e < 2). Then we find the
inequalities
hold with probability 1 only for finitely many n. However,
/ e\ xl2 1 — e ( ( e\1/'2\Sn > A J 6� and M~ > I 1 — A 1 I b~l i.o. a.s.
The above two statements also follow from Strassen's theorem 1 (cf. Section 8.1).Further,
Sn>(l--) b-1 and M" <n1/2(logn)-"/2 i.o. a.s. E.11)
if and only if rj < e.
NON-ACTIVATEDVERSIONwww.avs4you.com
Chapter 6
Wiener process and Invariance
Principle
6.1 Two lemmas
Clearly the r.v. a~1(S(k + a) — S(k)) can be considered as the average speed of
the particle in the interval [k,k + a). Similarly the r.v.
a-vIx{n,a) = a~x max {S{k + a) - S{k))
is the largest average speed of the particle in @, n) over the intervals of size a.
We know (Theorem 2.9) that a~xl'l{S{k + a) - S{k)) is asymptotically (a -> oo)an JV@,1) r.v. Hence S(k + a) - S(k) behaves like a1/2 or by the LIL of Khinchine
S(k + a) - S(k)hmsup-^— L
tttt"= ! a-s-
a^oo Ba log log aI/2for any fixed k. We prove that even Ii(n, a) cannot be much bigger than a1/2. In
fact we have
LEMMA 6.1 Let a = an < na @ < a < 1). Then
limsup , )'
. < C a.s.
n—oo na/2(lognI/2if C > 4.
Proof. By Theorem 2.10 for any k
P
(c2\
logn) = n"c2/2
47
NON-ACTIVATEDVERSIONwww.avs4you.com
48 CHAPTER 6
as n —+ oo. Hence
f )
and the Borel - Cantelli lemma implies the statement.
Remark 1. Much stronger results than that of Lemma 6.1 can be found in
Section 7.3.
LEMMA 6.2 Let {XfJ; * = 1,2,...;/ = 1,2,...} be a double array of i.i.d.r.v. 's
with
EXfJ- = 0, EXj = 1, Eexp(*XfJ) < oo
for all t in some interval \ t |< tQ. Then for any K > 0 there exists a positiveconstant C = C(K) such that
for all but finitely many i.
Proof of Lemma 6.2 is essentially the same as that of Lemma 6.1 using Theorem
2.11 instead of Theorem 2.10.
6.2 Definition of the Wiener process
The random walk is not a very realistic model of the Brownian motion. In fact the
assumption, that the particle goes at least one unit in a direction before turningback, is hardly satisfied by the real Brownian motion. In a more realistic model
of the Brownian motion the particle makes instantaneous steps to the right or to
the left, that is a continuous time scale is used instead of a discrete one.
In order to build up such a model assume that in a first experiment we
can only observe the particle when it is located in integer points and further
experiments describe the path of the particle between integers. Let
{S(n) = SW(n), n = 0,1,2,...}
be the random walk which describes the location of the particle when it hits
integer points. Then we define a new Brownian motion S^^n) which is a "re-
"refinement" of S^(n), i.e. S^^n) is a random walk with the properties:
(i) the particle moves 1/2 to the right or to the left with probability 1/2,
NON-ACTIVATEDVERSIONwww.avs4you.com
WIENER PROCESS AND INVARIANCE PRINCIPLE 49
(ii) the time needed for one step is 1/4,
(iii) S^(n) hits the integers in the same order as S^(n) does.
Observe that for S^(n) and S^(n) the expectations of the waiting times to hit
a given integer are equal to each other.
In order to construct a random walk which satisfies the above three conditions
let
{Sk{n) = Sk>1 (n), n = 0,1,2,...} {k = 1,2,...)be a sequence of independent random walks. Sk(n) governs the moving of the
particle between S(k — 1) and S(k).Introduce the following notations:
To = fo.i = 0,
rk = rkA = inf{j : |Sfc(y)| =2} [k = 1,2,...),k
Tk = Tktl = Y,n,i {k = 0,1,2,...),
ak = sign^S^) - S{k - l))Sk{Tk - Tk^)),
S{1) (^) = S{k - 1) + l-akSk{n - Tk.x) if TM < n < Tk {k = 1,2,...).
Observe that
where
A) = iand t0 = 0, tj, t2,... is a sequence of integers with \ix — io\ — |t2 — *i||»n+i — t'n| = 1. In other words
i.e. the finite dimensional distributions of S^(n/4) are equal to the correspond-corresponding distributions of S(n)/2. This result can be formulated as follows: S^(n/4)is a Brownian motion with the property that the particle moves 1/2 to the rightor to the left with probability 1/2 and the time needed for one step is 1/4. Since
(* = 0,1,2,...)
NON-ACTIVATEDVERSIONwww.avs4you.com
50 CHAPTER 6
we say that S^ is a refinement of
We define similarly the refinement S^(n/16) of S'A)(n/4). Hence we define
a sequence {Sk,2{n),n = 0>l>2,...} of independent random walks, being also
independent from the previously defined random walks. Now let
if 7*_i,j < n < r4i2 (k = 1,2,...) where
at" = sign ((sC> g) - S<" (^)) SW(T« - 7i-,,)) ,
A:
T^.2 = ?>,i2 (A; = 0,1,2,...),i=o
7-0,2 = 0,
It can be easily seen that
and
Hence we say that S^(n/16) is a Brownian motion with the property that the
particle moves 1/4 to the right or to the left with probability 1/2 and the time
needed for one step is 1/16. Further, S^ is a refinement of S^\
Continuing this procedure we obtain a sequence of random walks
as follows: having {$(m> (n2�m), n = 0,1,2,...} defined, (S'(m+1)(n2-2m-2), n =
0,1,2,...} will be denned by:
_ Mm) (k ~ l\ ,L>)c (T \
if Tk-i,m+i <n< Tk,m+i (fc = 1,2,...) where
4"> = sign ((s<"> (A] _ S(~) fci)Vk
Tk,m+l = ^Ti.m+l (fc = 0,1,2, ...),1=0
T0,m+l = 0,
ri,m+i = inf{j : |5«,m+i(i)| =2} (/ = 1,2,...)
NON-ACTIVATEDVERSIONwww.avs4you.com
WIENER PROCESS AND INVARIANCE PRINCIPLE 51
and {SiiTn+i(n),n = 0,1,2,...} (/ = l,2,...;m = 1,2,...) is a double array of
independent random walks.
It can be easily seen again that
and
= S<'V22-+2; \22-;-
Hence we say that S(m+1)(n2"�m~2) is a Brownian motion with the property that
the particle moves 2~(m+1) to the right or to the left with probability 1/2 and
the time needed for one step is 2"Bm+2). Further, S'(m+1) is a refinement of S^m\
A simple calculation givesEr,,w = 4,
and for any t0 > 0 there exists a C = C(t0) such that
Eexp(*r«im) <C
if \t |< t0.
Applying Lemmas 6.1 and 6.2 we find that for any K > 0 there exists a
C = C(K) > 0 such that
sup |S'(m+1)(A;2-2m) - S'(m)(A;2-2m)| < Cm2"m/2 a.s.
k2~'2m<K
for all but finitely many m. Hence as m —> oo the sequence
(A;2�m < t < (k + lJ�m) converges uniformly in any finite interval [0, T] to a
continuous function W(t) = W(t,u) for almost all u ? ft. This limit process is
called a Wiener process.
It is also easy to see that this limit process has the following three properties:
(i) W(t) - W(s) e N{Q,t- s) for all 0 < s < t < oo and W{0) = 0,
(ii) W{i) is an independent increment process that is ) )W(t4) - W(t3),..., W(t2i) -W{t2i.x) are independent r.v.'s for all 0 < tx <
ti <t3<U<...<t2i.1 <t2i (t = 2,3,...),
(iii) the sample path function W(t,u) is continuous in t with probability 1.
(i) and (ii) are simple consequences of the central limit theorem (Theorem 2.9).(iii) was proved above.
NON-ACTIVATEDVERSIONwww.avs4you.com
52 CHAPTER 6
6.3 Invariance Principle
Define the sequence of r.v.'s 0 < rt < r2 < ... as follows:
i-! =inf{t:t>O,\W{t)\ = l},r2 = mf{t : t > r^lWit) -WirJl = 1},
ri+l = \nf{t :t > r^lWit) -W(rt)\ = 1},...
Observe that
CO ^{ri),W{T2) - W{tx),W{tz) - W{t2),... is a sequence of i.i.d.r.v.'s with
distribution P{V^(r1) = 1} = TiW^) = -1} = 1/2, i.e. {W{Tn)} is a
random walk,
(ii) ri» r2~~
T\, Tz—
r2,... is a sequence of i.i.d.r.v.'s with distribution
Applying the reflection principle (formulated for Sn in Section 2.1, Proof 2 of
B.5)) for W(-) we obtain
'* °U~
) dt. F.1)
Evaluating the moments of rt and applying the strong law of large numbers we
obtain
Er! = 1, Er.2 = 2, lim n"Vn = 1 a.s.1
n—»oo
The above two observations are special cases of a theorem of Skorohod A961).Because of (i) we say that a random walk can be embedded to a Wiener process
(by the Skorohod embedding scheme).Applying the LIL of Hartmann - Wintner A941) (Section 4.4) and some
elementary properties of the Wiener process (formulated in Section 6.1) we obtain
\W(rn)-W(n)\hmSUp —
r-rr-T- -rjr < OO a.S.n-oo (n log log n) V4( log n) V2
This result can be formulated as follows:
THEOREM 6.1 On a rich enough probability space {fi, /, P} one can definea Wiener process {W(t),t > 0} and a random walk {Sn, n = 0,1,2,...} such that
\Sn-W(n)\hmsup j—— —j-t- —T- < oo a.s.
n-oo (n log log nI/4(log nI'2
NON-ACTIVATEDVERSIONwww.avs4you.com
WIENER PROCESS AND INVARIANCE PRINCIPLE 53
This result is a special case of a theorem of Strassen A964).A much stronger result was obtained by Komlos - Major - Tusnady A975-76).
A special case of their theorem runs as follows:
INVARIANCE PRINCIPLE 1 On a rich enough probability space {ft, /,P}one can define a Wiener process {W(t),t > 0} and a random walk {Sn, n =
0,1,2,...} such that
\Sn-W{n)\ = O{logn) a.s.
Remark 1. A theorem of Bartfai A966) and Erdos - Renyi A970) impliesthat the Invariance Principle 1 gives the best possible rate. In fact if {Sn, n =
0,1,2,...} and {W(t),t > 0} are living on the same probability space {fi, /,P}then | Sn - W(n) |> O(logn) a.s.
As a trivial consequence of Invariance Principle 1 we obtain
THEOREM 6.2 Any of the EKFP LIL, the Theorems of Chung and Hirsch
and Theorems 5.3, 5.4 and 5.5 remain valid replacing the random walk Sn bya Wiener process W(t). As an example we mention: Let Y(t) be any of the
processes
rl'*\w[t)\,t~Xl2m(t\ — t~Xl2 <;iin IWMI6 TJX 16 1 — 6 otl^J I v /1 ?
Q<s<t
t~l/2rn+(t) = t~1/2 sup WE),Q<s<t
+-1/2rn~(t) — —t~xl2 Jnf W( <i\L TJX 16 1 ——
6 1111 ww loliv '
0<$<tv '
Then a nondecreasing function a(t) € V\JC(Y(t)) if and only if
r°° a(t) ( a2(t)\I exp I I at < oo. ("-^JJ\ t \ 2 )
Similarly a nonincreasing function [a{t))~x € \AjC{t~xl2m{t)) if and only if
\dt<oc. F.3)
Remark 2. In fact the EFKP LIL only implies that a(n) e UUC(F(n)) (n =
1,2,...) if a(-) is nondecreasing and F.2) is satisfied. In order to get our Theorem
6.2 completely we have to know something about the continuity of W(-), i.e.
NON-ACTIVATEDVERSIONwww.avs4you.com
54 CHAPTER 6
we have to see that the fluctuation supfc<tsupJk<J<Jk+1 \W(s) — W(k)\ cannot be
very big. For example, the complete result can be obtained by Theorem 7.13,especially Example 2 of Section 7.2, which says that the above fluctuation is
asymptotically Blog*I/2 a.s.
Theorem 6.2 claimed that any of the strong theorems formulated up to now
for Sn will be valid for W(-). The same is true for the limit distribution theorems
of Section 2.2. In fact we have
THEOREM 6.3
^ u} = 2A - $(u)) (* > 0,u > 0),
? (-l)*exP U^f^l] dx1
Jfc=-oo
For later references we give a more general form of the Invariance Principle1:
INVARIANCE PRINCIPLE 2 (Komlos - Major - Tusnady, 1975-76). Let
F(x) be a distribution function with
f°° xdF(x) = 0, f°° x2dF{x) = 1— oo J—oo
'
etxdF(x) < oo | t \< t0
with some t0 > 0. Then, on a rich enough probability space {17,/,P}, one can
define a Wiener process {W(t),t > 0} and a sequence of i.i.d.r.v.'s Y\, Y2,... with
< x} = F(x) such that
\Tn - W(n)\=O(log n) a.s.,
where Tn = Yx + Y2 + • • • + Yn.
NON-ACTIVATEDVERSIONwww.avs4you.com
Chapter 7
Increments
7.1 Long head-runs
In connection with a teaching experiment in mathematics, T. Varga posed a
problem. The experiment goes like this: his class of secondary school children is
divided into two sections.
In one of the sections each child is given a coin which he then throws two
hundred times, recording the resulting head and tail sequence on a piece of paper.
In the other section the children do not receive coins but are told instead that
they should try to write down a "random" head and tail sequence of lengthtwo hundred. Collecting these slips of paper, he then tries to subdivide them
into their original groups. Most of the time he succeeds quite well. His secret
is that he had observed that in a randomly produced sequence of length two
hundred, there are, say, head-runs of length seven. On the other hand, he had
also observed that most of those children who were to write down an imaginaryrandom sequence are usually afraid of putting down head-runs of longer than
four. Hence, in order to find the slips coming from the coin tossing group, he
simply selects the ones which contain head-runs longer than five.
This experiment led T. Varga to ask: What is the length of the longest run
of pure heads in n Bernoulli trials?
A trivial answer of this question is
THEOREM 7.1
lim :—— = 1 a.s.N->oo lg N
where Zn is the length of longest head-run till N.
55
NON-ACTIVATEDVERSIONwww.avs4you.com
56 CHAPTER 7
Proof.
Step 1. We prove that
Iiminf-%>1 a.s. G.1)Noo lgjV
~
Let e < 1 be any positive number and introduce the notations:
t=[(l-?)\gN],
Uk = St(k+1) — Stk (k — 0,1
Clearly UQ, U1,..., Ujf are i.i.d.r.v's with
1
p(^ = t) = —.
2*
Hence
i>(uQ<t,ul<t,...,uN-<t) = (i-^and a simple calculation gives
~N
-I < OO
for any ? > 0. Now the Borel - Cantelli lemma implies G.1).
Step 2. We prove that
limsup:—^r<l a.s. G.2)N—•• Ik N
Let ? be any positive number and introduce the following notations:
u = [(l +
fc = Sk+U-Sk {k = 0,l,...,N-u
= \J{Vk = u}Jk=O
and let T be any positive integer for which Te > 1. Then
P(Vt = u) = 2"u,
consequentlyoo
) < N2~u and
NON-ACTIVATEDVERSIONwww.avs4you.com
INCREMENTS 57
Hence the Borel - Cantelli lemma implies
ZuTlimsup—-F < 1 a.s. G.3)
jk^oo \gkT '
Let kT < n < (k + l)T and observe that by G.3)
Zn < Z{k+l}r <{! + ?) \g{k + 1)T < A + 2e) \gkT < A + 2e) lgn
with probability 1 for all but finitely many n. Hence we have G.2) as well as
Theorem 7.1.
A much stronger statement is the following:
THEOREM 7.2 (Erdos - Revesz, 1976). Let {an} be a sequence of positivenumbers and let
n=l
Then
an € UUC(Zn) if A{{an}) < oo, G.4)an G ULC(Zn) if A{{an}) =00 G.5)
and for any e > 0
An = [lgn - lglglgn + lglge - 2 - e] e LLC(Zn). G.7)
Example 1. If 6 > 0 and
< = lg n + A + <5) lg lg n then A({<}) < 00.
Hence G.4) and G.7) together say that
\n< Zn< a*n for all but finitely many n} = 1.
Note that if n = 223° = 21048576 ~ 10315621 and e = 6 = 0,1 then An = 1048569
and < = 1048598.
Remark 1. Clearly G.4) and G.5) are the best possible results while G.6) and
G.7) are nearly the best possible ones.
A complete characterization of the lower classes was obtained by Guibas - OdlyzkoA980) and Samarova A981). Their result is:
NON-ACTIVATEDVERSIONwww.avs4you.com
58 CHAPTER 7
THEOREM 7.3 Let
i^n = ^gn - \g\g\gn + \g\ge - 2.
Then
liminf[Zn - xjjn] = 0 a.s.n—»oo
It is also interesting to ask what the length is of the longest run containingat most one (or at most T, T = 1,2,...) (-l)'s. Let Zn(T) be the largest integerfor which
Il(n,Zn(T))>Zn(T)-2T
where
/i(n, a) = max {Sk+a - Sk).
A generalization of Theorem 7.2 is the following:
THEOREM 7.4 Let {an} be a sequence of positive numbers and let
n=l
Then
ane\J\JC{Zn{T)) if AT{{an}) < oo, G.8)an e ULC(Zn(T)) if ^T({an}) = oo G.9)
and for any e > 0
Kn{T) = [lgn + Tlglgn - lglglgn - lgT! + lglge - 1 + e]G.10)
Xn{T) = [lgn + Tig lgn- lglglgn- lgT'. + lglge- 2- e]eLLC(Zn(T)). G.11)
A trivial reformulation of the question of T. Varga is: how many flips are needed
in order to get a run of heads of size m? Formally speaking let Zm be the smallest
integer for which
As a trivial consequence of Theorem 7.2 we obtain
NON-ACTIVATEDVERSIONwww.avs4you.com
INCREMENTS 59
THEOREM 7.5
Am € UUC(Zm),Km e ULC(Zm),am e LUC(Zm) if A{{am}) = oo,
am e LLC(Zm) if A{{am}) < oo
where Km resp. Xm are the inverse functions of Km of G.6) resp. Xm of G.7) and
am is the inverse function of the positive increasing function dm.
Instead of considering the pure head-runs of size m one can consider any givenrun of size m and investigate the waiting time till that given run would occur.
This question was studied by Guibas - Odlyzko A980).Erdos asked about the waiting time Vm till all of the possible 2m patterns of
size m would occur at least once. An answer of this question was obtained byErdos and Chen A988). They proved
THEOREM 7.6 For any e > 0
lge
and
?)r ¦- e uuc(vm)mm
A much stronger version of this theorem was obtained by Mori A989). He proved
THEOREM 7.7 For any e > 0
)-1 e UUC(Vt),)-1 e ULC(Vt),
{2kk - {1 - ?Jklglgk){lge)-1 e LVC{Vk),{2kk - {1 + sJk\g\gk){\ge)-1 e LLC{Vk).
We mention that the proof of Theorem 7.7 is based on the following limit distri-
distribution:
THEOREM 7.8 (Mori, 1989).
lim 2*/18sup {2~kVk -
{< y) - e~e = 0.
NON-ACTIVATEDVERSIONwww.avs4you.com
60 CHAPTER 7
In order to compare Theorem 7.7 and Theorems 7.2 and 7.3 it is worthwhile to
consider the inverse of Vk. Let
Un = max{k : Vk < n).
Then Theorem 7.7 implies
Corollary 1. (Mori, 1989). For any ? > 0 we have
[lgn-lglgn-elge^^j < Un < flgn - lglgn + (l + e) lge^-^l a.s.
[ lgn J L 18n J
for all but finitely many n. Consequently Un is QAD.
Observe that Un is "less random" than Zn. In fact for some n's the lower and
upper estimates of Un are equal to each other and for the other n's they differ
by 1. Clearly Un < Zn but comparing Theorems 7.2, 7.3 and Corollary 1 it turns
out that Un is not much smaller than Zn.In Theorem 7.2 we have seen that for all n, big enough, there exists a block of
size An (of G.7)) containing only heads but it is not true with Kn (of G.6)). Now
we ask what the number is of disjoint blocks of size An containing only heads.
Let un(k) be the number of disjoint blocks of size k (in the interval [0, n])containing only heads, that is to say un(k) = j if there exists a sequence 0 < ti <
ti + k < t2 < t-i + k < ... < tj < tj + k < n such that
Stt+k-St! = k (i = 1,2 j)
but
Sm+k ~Sm<k if U + k < m < ti+l (i = 1,2,... ,j - 1)
or tj + k < m < n — k.
The proof of the following theorem is very simple.
THEOREM 7.9 (Revesz, 1978). For any e > 0 there exist constants 0 < ax =
ai{?) 5: <*2 = Q=2(e) < oo such that
= hminf -—;—- < hmsup t—: = a2 a.s.n—oo lglgn n—oo lglgn
{for Xn see G.7)).
This theorem says that in the interval [0, n] there are O(lg lg n) blocks of size Ancontaining only heads. This fact is quite surprising knowing that it happens for
NON-ACTIVATEDVERSIONwww.avs4you.com
INCREMENTS 61
infinitely many n that there is not any block of size An + 2 > Kn containing onlyheads.
Deheuvels A985) worked out a method to find some estimates of ai(e) and
a~{s). In order to formulate his results let Zn = Z^ and let Z™ >Z^]>... be
the length of the second, third, ... longest run of l's observed in Xi, X2,..., Xn.Then
THEOREM 7.10 (Deheuvels, 1985). For any integer r > 3 and
k > 1 and for any e > 0
G.12)
(lg2n+ + lgr_1n + lgrn) G ULC(Z), G.13)
[lgn-lg3n + lglge-l]eLUC(ZW), G.14)
[lgn - lg3n + lglge - 2 - e] G LLC(zW). G.15)
Remark 2. In case k = 1 G.14) gives a stronger result than G.6) but G.14)and G.15) together is not as strong as Theorem 7.3.
THEOREM 7.11 (Deheuvels, 1985). Let v e @,+oo) be given, and let 0 <
c[ < 1 < c" < oo be solutions of the equation
c-l-logc = -. G.16)
Then for any e > 0 we have
^1l G.17)G.18)
[lgn - lg3 n + lg2 e - \gc'v + e\ G LUC(Zi"lo«»Bl), G.19)[lgn - lg3 n + lg2 e - lg< - 2 - e) G LLC(Z^°*^). G.20)
Remark 3. This result is a modified version of the original form of the theorem.
It is also due to Deheuvels (oral communication).Theorem 7.2 also implies that
liminf i/n(/n) = 0 a.s.
if ln > Xn but
.. ...
,,... .
n f = 0 if 6 > 0,limsupi/n([lgn+(H-«)lglgn])< f
.
Now we are interested in limsup,,^^ ^n([lg n + lg lg n]) an(l formulate our simple
NON-ACTIVATEDVERSIONwww.avs4you.com
62 CHAPTER 7
THEOREM 7.12 (Revesz, 1978).
limsupi/n([lgn +lglgn]) < 2 a.s. G.21)n—»oo
Finally we mention a few unsolved problems (Erdos - Revesz, 1987).
Problem 1. We ask about the properties of Zn - Z& = Z^ - Z?\ It is clear
that P{ZW = zW i.o. ) = 1. The limsup properties of Z^ - Z& look harder.
Problem 2. Let Kn be the largest integer for which
Characterize the limit properties of Kn. Observe that Theorem 7.9 suggests
0 < limsup -—: < oo.
n—oo log log U
Problem 3. Let Z^ be the length of the longest tail run, i.e. Z^ is the largestinteger for which
where
r(n,k)=Q<mm_k(Sj+k-S}).How can we characterize the limit properties of \Zn — Z^\l Note that by Theorem
7.2
limsup —— < 1 a.s.
n—oo log log Tl
and clearlyP{Zn = Z*n i.o.} = 1.
_|0 if Zn<Z*n,Un 11 if Zn>Z*n
Problem 4. Let
and
i.e. Un = 1 if the longest head run up to n is longer than the longest tail run. We
ask: does lim,,.^ ?n exist with probability 1? In the case when \im.n-tooXn = ?a.s. then ?. is called the logarithmic density of the sequence {Un}.
NON-ACTIVATEDVERSIONwww.avs4you.com
INCREMENTS 63
Problem 5. (Karlin - Ost, 1988). Consider two independent coin tossingsequences Xu X2,..., Xn and X[, X2,..., X'n. Let Yn be the longest common
"word" in these sequences, i.e. Yn is the largest integer for which there exist a
1 < kn< kn + Yn<n and a 1 < k'n < k'n + Yn < n such that
Xkn+j = Xk'r.+i if i = 1> 2, • • •, Vn-
Karlin and Ost A988) evaluated the limit distribution of Yn. Its strong behaviour
is unknown. Petrov A965) and Nemetz and Kusolitsch A982) investigated the
length of the longest common word located in the same place, i.e. they defined
Yn assuming that kn = k'n. In this case they proved a strong law for Yn.
7.2 The increments of a Wiener process
This paragraph is devoted to studying the limit properties of the processes
Mt,at)(i = 1,2,3,4,5) where at is a regular enough function (cf. Notations to the
increments).Note that the r.v. a~1(W(s + a) — W(s)) can be considered as the average
speed of the particle in the interval (s, s + a). Similarly the r.v.
a-1J1{t,a)=aT1 sup (W{s + a) - W{s))0<3<t-a
is the largest average speed of the particle in @, t) over the intervals of size a.
The processes J,(t,a)(i = 2,3,4,5,? > a) have similar meanings.Note also that
Ji{t,at) < min{J2{t,at),J3{t,at)}max{J2{t,at),J3(t,at)} < J4{t,at).
To start with we present our
THEOREM 7.13 (Csorgo - Revesz, 1979/A). Let at(t > 0) be a nondecreasingfunction of t for which
(i) 0 < at < t,
(ii) t/at is nondecreasing.
Then for any i = 1,2,3,4 we have
at) — W(t)\()t—»oo t—»oo
at) - W{t)) = 1 a.s.
t-*oo
NON-ACTIVATEDVERSIONwww.avs4you.com
64 CHAPTER 7
where
= [2at (log— + log log t
at
If we have also
(iii)lim (log — ) (log log t)
1=
t-K» \ at/
= oo
then
lim itJi(t,at) = 1 a.s.t—»oo
In order to see the meaning of this theorem we present a few examples.
Example 1. For any c > 0 we have
lf a.s. A = 1,2,3,4). G.22)t
This statement is also a consequence of the Erdos - Renyi A970) law of largenumbers.
Example 2.
lim ,:?_'*{l9 = 1 a.s. (x = 1,2,3,4). G.23)
Example 3. For any 0 < c < 1
In case c = 1 we obtain the LIL for Wiener process (cf. Theorem 6.2). Note that
G.24) is also a consequence of Strassen's theorem of 8.1.
Having Theorem 7.13 it looks an interesting question to describe the Levy-classes of the processes J,-(t,at)(t = 1,2,3,4) in case of different at's. Unfortu-
Unfortunately we do not have a complete description of the required Levy-classes. We
can only present the following results:
THEOREM 7.14 (Ortega- Wschebor, 1984). Let f(t) be a continuous nonde-
creasing function and assume that at satisfies conditions (i) and (ii) of Theorem
7.13. Then
f{t) e UUC (ar1/2J,.(*,at)) (r= 1,2,3,4)
NON-ACTIVATEDVERSIONwww.avs4you.com
INCREMENTS 65
2
Li?i«p _O!2 *<«». G.25)<*t \ 2 J
Further, if
I ^-^-exp I — I dt = oo G.26)
f{t) e ULC (ar1/2Ji(t,at)) (i = 1,2,3,4).
Remark 1. In case at = t condition G.26) is equivalent with the correspondingcondition of the EFKP LIL of Section 5.2. However, condition G.25) does not
produce the correct UUC in case at = t. Hence it is natural to conjecture that,in general, the UUC can be characterized by the convergence of the integral of
G.26). It turns out that this conjecture is not exactly true. In fact Grill A989)obtained the exact description of the upper classes under some weak regularityconditions on at. He proved
THEOREM 7.15 Assume that
at = Coexp( T —dy) < St
where 0 < 6 < 1, g(y) is a slowly varying function as y —> oo,Co,C1 are posi-positive constants.
Let f(t) > 0(t > 0) be a nondecreasing function. Then
f{t) e UUC (ar1/2(J,(*,at)) (*' = 1,2,3,4)
if and only if
< oo.
In order to illustrate the meaning of this theorem we present a few examples.
Example 4. Let at = {\ogt)a (a > 0). Then g(t) = a/log* and
( "-1 V"fp<e{t) = 2 log* + C - 2a) log21 + 2 ? logj. t + B + e) logp t
e UUC (ot�/2J,) if and only if ? > 0 (i = 1,2,3,4; p = 3,4,5,...).
NON-ACTIVATEDVERSIONwww.avs4you.com
66 CHAPTER 7
Example 5. Let at = exp((logOa) @ < a < 1). Then g(t) = a(logO"� and
/p,,@ = 2 log t - 2(log t)a + C + 2a) log21 + 2 ? log, t + B + e) logp t
e UUC (at�/2J,) if and only if ? > 0 (i = 1,2,3,4; p = 3,4,5,...).
Example 6. Let at = ta@ < a < 1). Then </(«) = a and
/p,«@ = I 2A - a) log t + 5 log21 + 2 ^ log, * + B + e) logp t
e UUC (at~1/2 Ji) if and only if e > 0 (i = 1,2,3,4; p = 3,4,5,...).
Example 7. Let at = at @ < a < 1). Then </(«) = 1 and
/ P-i
/p,«@ = 2 log21 + 5 log31 + 2 ? log, t + B + e) logpV i=*
e UUC (aTl/2J,-) if and only if e > 0 (i = l,2,3,4;p = 4,5,6,...).
Theorem 7.15 does not cover the case at/t —> 1. As far as this case is concerned
we present
THEOREM 7.16 (Grill, 1989). Let at = t(l - 0(t)) where 0{t) is decreasingto 0 and slowly varying as t —> oo and f(t) > 0 be a nondecreasing continuous
function. Then
f{t) e UUC (aJ^J^at)) {i = 1,2,3,4)
if and only if
< oo.
The characterization of the lower classes is even harder. At first we present a
theorem giving a nearly exact characterization of the lower classes when at is not
very big.
THEOREM 7.17 (Grill, 1989). Assume that
T
NON-ACTIVATEDVERSIONwww.avs4you.com
/\ct y
with(,. log*7(* . .
hmsup-——5-L < -«t-oo log log t
or
log log t/at_^
log log t
then
K1 = \ogn, K2 = \og-,4
log- < K3 < log7T,4
INCREMENTS 67
Then for any i — 1,2,3,4 we have
LUL. iCtj Jt[t, (It) j 1/ A < A,-
where
logTr < Ki < Iog47r,
log - < K-i < log it,47T
log- < ^3 < log47T,4
log — < ^4 < log 7T.
lb
// in addition either at is of the form
at = Coexp
^Kilogj.16 4
Remark 2. A very similar result was obtained previously by Revesz A982).However, some of the constants given there are not correct.
Example 8. Let at = «e-rlo«logt@ < r < oo). Then
A(t) = (exp(rloglogO)(loglog«)� T oo.
Hence.. .
c Ji{t,at)hminf t :—; tttt
= 1 a.s.t-oo Batrloglog«I/2
This result was proved by Book and Shore A978).If at is so big that the condition A(t) j oo does not hold the situation is even
more complicated. We have two special results (Theorems 7.18 and 7.19) only.
NON-ACTIVATEDVERSIONwww.avs4you.com
68 CHAPTER 7
THEOREM 7.18 (Csaki - Revesz, 1979, Grill, 1989). // A{t) = C > 0, i.e.
at = Ct(loglogt)� then with probability 1 we have
...... . / +oo if C < T,hminfJ1(*,at) =
j_oo lf C>T
where T is an absolute, positive constant, its exact value unknown.
If A(t) ->() then
lim inf —p , , =-/?( — ) a.s.«-« ^24 log log * V t /
and
Remark 3. Note that if at = at and I/a is integer then fi(at/t) = a. We
return to the discussion of this theorem in Section 8.1 in the special case when
at = at @ < a < l). The first part of Theorem 7.18 suggests the followingquestion. Does there exist a function at for which liminfj-^oo Ji(t,at) = 0 a.s.?
THEOREM 7.19 (Csaki - Revesz, 1979). // A{t) — 0 then
1 < liminf <5(«) J4{t,at) < 2>/2 a.s.t—*OO
where
(*2 r1/26(t) = [jJ
Remark 4. In case at = t, Theorem of Chung (Section 5.3) implies that
liminf 6(t)J4(t,at) = 1 a.s.
However, this relation does not follow from Theorem 7.19.
Remark 5. Ortega and Wschebor A984) also investigated the upper classes of
the "small increments" ofW(-). These are defined as follows:
Mt,at)= sup {W{s + as)-W{s)),0<3<t-a,
NON-ACTIVATEDVERSIONwww.avs4you.com
INCREMENTS 69
J7{t,at)= sup \W{s + a3)-W{s)\,0<s<t-at
JB(t,at) = sup sup (W(s + u)-0<s<t-at 0<u<a,
J9(t,at) = sup sup \W(s + u) - W(s)\0<s<t-a, 0<u<a.
where a, is a function satisfying conditions (i) and (ii) of Theorem 7.13.
Remark 6. Hanson and Russo A983/B) studied a strongly generalized version
of the questions of the present paragraph. In fact they described the limit pointsof the sequence
W{pk)-W{ak)B@k - ak)(\og@k/@k - ak)) +
for a large class of the sequences 0 < ak < /3k < oo.
Finally we present a result on the behaviour of J${-, •)¦
THEOREM 7.20 (Csorgo - Revesz, 1979/B). Assume that at satisfies condi-
conditions (i), (ii) of Theorem 7.13. Then
where
and
liminf Kt Js(t,at) = 1 a.s.t—»oo
Js{t,at)= inf sup \W(s + u)-W{s)Q<s<t-at o<u<at
'"I"// (iii) of Theorem 7.13 is also satisfied then
lim KtJs{t,at) = 1 a.s.t—>OO
The following examples illustrate what this theorem is all about.
Example 9. Let at = ^\ogt hence Kt -»• l{t -> oo). Then Theorem 7.20 tellsus that for all t big enough, for any e > 0 and for almost all a; € ft there existsa 0 < s = s{t,?,u) < t- at such that
sup \W(s + u) -W(s)\ < 1 + e
NON-ACTIVATEDVERSIONwww.avs4you.com
70 CHAPTER 7
but, for all s G [0,t — at\ with probability 1
sup \W{s + u)-W(s)\>l-?.
At the same time Theorem 7.13 stated the existence of an s G [0, t — at\ for which,with probability 1,
s+^-Xogt} -W{s)
but for all s e [0,t - at\
sup \W{s + u) -W{s) | <(- + e) log t.
Example 10. Let at = t. Then Theorem 7.20 implies
\1/2su ^t—oo y nH J 0<3<t
Hence we have the Other LIL (cf. E.9)).
Example 11. Let at = (log*I/2 hence Kt « ^(log01/4- Then Theorem 7.20
claims that for all t big enough, for any e > 0 and for almost all u G ft there
exists an s = s(t, e,uj) G [0, t — at] such that
supo<u<(iogtI/2
That is to say the interval [0,t — at] has a subinterval of length (logfI/2 where
the sample function of the Wiener process is nearly constant; more precisely, the
fluctuation from a constant is as small as A + ?Or8~1/2(logt)~1/4.This result is sharp in the sense that for all t big enough and all s ? [0, t — at]
we have with probability 1
sup \W{s + u) - WE)| > A - ?)^=(log0-1/4.0<u<(logtI/2 V°
Clearly, replacing the condition at = (log?I//2 in Example 11 by at = o(logt),we also find that there exists a subinterval of [0, t — at] of size at where the samplefunction is nearly constant. Csaki and Foldes A984/A) were interested in the
analogue problem when the term "nearly constant" is replaced by "nearly zero".
They proved
NON-ACTIVATEDVERSIONwww.avs4you.com
INCREMENTS 71
THEOREM 7.21 Assume that at satisfies conditions (i), (ii) of Theorem 7.13.
Then
liminf/it inf sup |W(s + u)| = 1 a.s.t-oo C0<5<t-a« o<u<a,
where
ir2at
If (iii) of Theorem 7.13 is also satisfied then
lim ht inf sup |W(s + u)| = l a.s.?-00 0<»<t-a«0<u|ac
' V "
Example 12. Letting at = t we obtain the Other LIL (cf. E.9)).
Example 13. If at = o(logt) then ht —> oo and
lim inf sup \W(s +u)| = 0t-ooO<J<t-a«0<u<a«
while in case at = 4c*7r~2 \ogt we have /it —» c� as ? —> oo and
lim inf sup \W(s + u)\ = c.
t-ooO<»<ta<|' V ;l
(Compare Example 13 in case c = 1 and c = \/2 with the first part of Example9.)
Theorems 7.13 - 7.19 gave a more or less complete description of the strongbehaviour of J,-(t, at)(i = 1,2,3,4). To complete this Section we give the followingweak law:
THEOREM 7.22 (Deheuvels - Revesz, 1987). Let t/at = dt. Assume that
lim dt = oo. G.27)
Then for any i = 1,2,3,4 in probability
G 2g)t—oo log log dt
We also mention that the proof of Theorem 7.22 is based on the following:
NON-ACTIVATEDVERSIONwww.avs4you.com
72 CHAPTER 7
THEOREM 7.23 (Deheuvels - Revesz, 1987). Assume G.27). Then for any
—oo < y < oo we have
—» exp(—e~v) (? —* oo)
t/ x = 2,3,4 and
-> exp(-e"») (t -> oo).
Note that in the above two theorems we have no regularity conditions on at
except G.27).In order to understand the meaning of G.28) consider the case t = 2 and
assume
-i^- = r @<r<oo). G.29)log log t
K ' K '
In this case Theorem 7.13 implies that J2{t, at) can be as big as
In the same case Theorem 7.17 implies that Ji{t,at) can be as small as
BatI/2(rloglog01/2.
G.28) describes the "typical" behaviour of J2(t,at) under the condition of G.29).Namely it behaves like
It is worthwhile to mention an equivalent but simpler form of Theorem 7.23.
THEOREM 7.24
limp] sup(WE + l)-WE)) <f{y,t)\ =exp{-e~v)t-*°° [0<3<t J
and
where
lim P I sup \W(s + l)-W (s) \<f(y,t)\= exp(-2e~v)
NON-ACTIVATEDVERSIONwww.avs4you.com
INCREMENTS 73
Let us give a summary of the results of this section. To study the properties of the
processes Ji(t,at) (i = 1,2,3,4,5) we have to assume different conditions on at.
For the sake of simplicity from now on we always assume that at is nondecreasingand satisfies conditions (i) and (ii) of Theorem 7.13.
Then the limit distributions of ./,(•,•) for i = 1,2,3,4 are given in Theorem
7.23. Observe that the limit distributions in case i = 1 and in case i = 2,3,4 are
different. The limit distribution of J5(-, •) is unknown. The exact distribution is
not known in any case.
A description of the upper classes of J,(-, •) (t = 1,2,3,4) is given in Theorem
7.14 but there is a big gap in this theorem between the description of UUC(J,) and
ULC(J,), i.e. there is a big class of very regular functions for which Theorem
7.14 does not tell us whether they belong to the UUC(J.) or to the ULC(J,).This gap is filled in by Theorem 7.15 if at satisfies a weak regularity condition.
However, this regularity condition excludes the case at/t —> 1. This case is
studied in Theorem 7.16. The above-mentioned results do not tell us anythingabout Js(-,-). In case if at is not very big (condition (iii) of Theorem 7.13 is
satisfied) a very weak result is given in Theorem 7.20.
The lower classes of J,(-, •) (i = 1,2,3,4) are "almost" completely described if
at <C tj log log t by Theorem 7.17. If at does not satisfy this condition Theorems
7.18, 7.19 resp. 7.20 tell something about the lower classes of Ji(-, •), J^-, •) resp.
Js(-, •) but we do not have a complete characterization and we do not have any
results (except trivial ones) about the lower classes of ^O,') and Js(-,-).
7.3 The increments of
By the Invariance Principle 1 (cf. Section 6.3) we obtain that any theorem of
Section 7.2 will remain true, replacing the Wiener process by a random walk (i.e.replacing J, by J,-(t = 1,2,3,4,5)) provided that '¦y� is big enough or equivalentlyan is big enough. In fact Theorems 7.13, 7.18 - 7.21 resp. 7.14 - 7.17 remain
true if an » logn resp. an » (lognK while Theorems 7.23 and 7.24 remain
true as they are. Hence we only study the increments of Sn in the case when
Hindoo an(logn)~3~e = 0 for any e > 0.
A trivial consequence of Theorem 7.1 resp. Theorem 7.13 (cf.also Example 1
of Section 7.2) is
THEOREM 7.25
lim/l("'lgn)=l a... G.30)n—oo lgn
NON-ACTIVATEDVERSIONwww.avs4you.com
74 CHAPTER 7
G.31)lgn n^°° logn
Remark 1. Comparing G.30) and G.31) we can see that the behaviours of 7iand Jx are different indeed if an = c logn@ < c < oo). As a consequence we also
obtain that the rate O(logn) of Invariance Principle 1 (cf. Section 6.3) is the
best possible one. This observation is due to Bartfai A966) and Erdos - Renyi
A970).Theorems 7.2, 7.3, 7.4 imply much stronger results on the behaviour of /,(-, •)
than G.30) of Theorem 7.25. In fact we obtain
THEOREM 7.26 Assuming different growing conditions on {an} we get
(i) // for some e > 0
on< [lgn-lglglgn + lglge-2-e] = An.
Then
Ii(n,an) = an a.s. (i = 1,2,3,4)
for all but finitely many n, i.e. 7,(n,an) is AD.
(ii) Let
An <an < [lgn + lglgn-lglglgn + lglge-2-<¦:] = An(l).
Then Ii(n,an)(i = 1,2,3,4) is QAD and 7,(n,an) = an or an— 2 a.s. for
all but finitely many n.
(iii) Let
An(l) <an< [lgn + lglgn + (l + e)lglglgn] = dn(l).Then 7,(n,an)(i = 1,2,3,4) is QAD and 7j(n,an) = an or an
— 2 or an— 4
a.s. for all but finitely many n.
(iv) In general, if
dn(T) = [lgn + Tig lgn + (l + e) lglglgn] < an < An(T + 1)= [lgn+(T+l)lglgn-lglglgn-lg((T + l)!) + lglge-2-e]
then 7j(n, an) is QAD and 7,(n, an) = an— 2T — 2 or an
— 2T, and if
An(T+l) <an<dn{T + l)
then 7,(n,an) is QAD and 7;(n,an) = an-2T-4 or an-2T-2 or an-2T.
NON-ACTIVATEDVERSIONwww.avs4you.com
INCREMENTS 75
The above theorem essentially tells us that Ii(n,an) is QAD with not more than
three possible values if an < lgn + Tiglgn for some T > 0. The next theorem
applies for somewhat larger an.
THEOREM 7.27 (Deheuvels - Erdos - Grill - Revesz, 1987). Let an = O(lgn)and 0 < T = Tan < an/2 be nondecreasing sequences of integers. Then
an-2TeLLC{Ii{n,an)) if ? exp(-2npBn)) < oo,n=l
an- 2T e LUC(/,(n, an)) if ? exp(-2npBn)) = oo,
n=l
an-2TeULC{Ii{n,an)) if f) 2>Bn) = oo,n=l
an -2TeUUC{It;(n,on)) t/ f) 2>Bn) < oo
n=l
tu/icrc
Here we present a few consequences.
Consequence 1. Let
an = lgn + /(n)be a nondecreasing sequence of positive integers with f(n) = o(lg n).
(i) Assume that
lim / ^n~= 0 for any e > 0.
n—oo (lgn)eThen 7,(n,an) is QAD and there exist an a.i(n) G UUC(/,(n,an)) and an
a4(n) G LLC(/,(n,an)) such that ai(n) — a^(n) < 3.
(ii) Assume that
/(n) = O((lgn)e) @<6<l).Then 7,(n,an) is QAD and there exist an a\{n) G UUC(/,(n,an)) and an
cn(n) G LLC(/,(n,an)) such that a^n) - a4(n) < ^ + 1.
(iii) Assume that
lim -—^— = oo for any e > 0.n~>0° (lgn)
Then /,(n,an) is not QAD.
NON-ACTIVATEDVERSIONwww.avs4you.com
76 CHAPTER 7
Consequence 2. Let an = [C lgn] with C > 1. Then
eJ/9lglgn€UUC(/i(n,an)),C(l-2/?)lgn+(l-eJplglgn€ ULC(/,(n,an)),
C(l - 2/3) lg n - 2p lg lg n - Ap lg lg lg n+
+ l + se LUC(/,(n,an)),
C(l-2/?)lgn-2plglgn-4plglglgn+4p lg(l - 2/3) + 4p lg lg c + 2p lg 7T + Gp + 1 - e € LLC^n, an))
where /? is the solution of the equation
I^)C = 2,
and e is an arbitrary positive number.
Remark 2. Consequence 2 above is a stronger version of an earlier result of
Deheuvels - Devroye - Lynch A986).In the case an » lg n we present the following:
THEOREM 7.28 (Deheuvels - Steinebach, 1987). Let a^ be a sequence ofpositive integers with an = [an] where an/logn is increasing and an(logn)~p is
decreasing for some p > 0. Then for any e > 0 we have
anan- tflogan + C/2 +*-)*;1 log log n G UUC(/,(n,an)),
anan- t~l logan + C/2 - e)t~l log log n € ULC(/,(n,an)),
anan-^1logan + (l/2 + e)^1loglogn € LUC(/,(n,an)),anan - t-1 \ogan + A/2 - e)t~l log log n € LLC(/,(n,an))
where an is the unique positive solution of the equation
exp(-(logn)/an) = A + an)A+
and1
,1 + an
tn = -log- .
2 1 -
an
Note that an » Ba;1 log nI/2.
NON-ACTIVATEDVERSIONwww.avs4you.com
INCREMENTS 77
In order to study the properties of 1$ resp.
lUn,an) = min max \Sj+i\,SK ' n)0<j<n-an 0<i<an
' } '
first we mention that by the Invariance Principle the properties of J$ resp.
j;(t,at) = inf sup0<«<«-a, o<u<at
will be inherited if an > (lognK+e(e: > 0). In fact Theorems 7.20 and 7.21 will
remain true if J5 resp. J5* are replaced by 75 resp./j and an > (lognK+e(e > 0).Hence we have to study the properties of 75 resp. /? only when an(log n)~3~e —>¦
0 (n —>¦ oo) for any e > 0. It turns out that Theorem 7.21 remains true if
an/ logn —>¦ oo (n —>¦ c»). In fact we have
THEOREM 7.29 (Csaki - Foldes, 1984/B). Assume that an satisfies condi-
conditions (i) and (ii) of Theorem 7.13 and
lim an(logn)� = oo.n—»oo
Then
lim inf hnlc(n,an) = 1 a.s.n—»oo
where hn is defined in Theorem 7.21. J/ condition (iii) o/ Theorem 7.13 is a/so
satisfied, then
lim /in/5*(n,an) = 1 a.s.n—»oo
If an = [c log n] then we have
THEOREM 7.30 (Csaki - Foldes, 1984/B). Let an = [c log n](c > 0) and definea* = a*(c) > 1 as the solution of the equation
if a*(c) is not an integer then
a.s.
for,all but finitely many n, i.e. II is AD,if a*(c) is an integer then
*{c)-l<r5{n,an)<a*{c) a.s.
for all but finitely many n, i.e. II is QAD. Moreover
75*(n,an) = a{c) - 1 i.o. a.s.
and
75*(n,an) = a*{c) i.o. a.s.
NON-ACTIVATEDVERSIONwww.avs4you.com
78 CHAPTER 7
The properties of 75 are unknown when log n <C an < (log nK. However, we have
THEOREM 7.31 (Csaki- F51des, 1984/B). Letan = [clogn\(c > 0) and definea = a(c) > 1 as the solution of the equation
¦K
cos -— =
if a(c) is not an integer then
h{n,an) = \a(c)} a.s.
for all but finitely many n, i.e. 1$ is AD,if a(c) is an integer then
a(c) — 1 < 75(n,an) < a(c) a.s.
for all but finitely many n, i.e. 1$ is QAD. Moreover
Is = a(c) — 1 i.o. a.s.
and
Is = a(c) i.o. a.s.
NON-ACTIVATEDVERSIONwww.avs4you.com
Chapter 8
Strassen type theorems
8.1 The theorem of Strassen
The Law of Iterated Logarithm of Khinchine (Section 4.4) implies that for any
e > 0 and for almost all uj ? Q there exists a random sequence of integers0 < i%i = rii{e,u>) <n-i — n2(e,u>) < ... such that
S{nk) > A -e)Bn4loglogn*I/2 = A - ej^n*))�. (8.1)
We ask what can be said about the sequence {Sj-,j = 1,2,..., nk} (provided that
(8.1) holds). In order to illuminate the meaning of this question we prove
THEOREM 8.1 Assume that nk = nk(e,u) satisfies (8.1). Then
S([nk/2}) > A - e)±S{n>) > ^^(ftK))� a.*. (8.2)
for all but finitely many k.
Proof. Let 0 < a < 1 — 2e and assume that
a^K))� < S{\nk/2\) < [a + e)[b{nk))-\ (8.3)
Then by (8.1)S(nk) - S([nk/2}) > (l - a - 2e)(b(nk))-\ (8.4)
By Theorem 2.10 the probability that the inequalities (8.3) and (8.4) simultane-
simultaneously hold is equal to O((lognik)-2(a2+A-a-2eJ)).Observe that if a / 1/2 and e is small enough then 2(a2 + A - a — 2eJ) > 1.
Hence by the method used in the proof of Khinchine's theorem (Step 1) we obtain
79
NON-ACTIVATEDVERSIONwww.avs4you.com
80 CHAPTER 8
that the inequalities (8.3) and (8.4) will be satisfied only for finitely many k with
probability 1. This fact easily implies Theorem 8.1.
Similarly one can prove that for any 0 < x < 1
S{[xnk})>{l-e)xS{nk) a.s. (8.5)
for all but finitely many k.
(8.5) suggests that if nk satisfies (8.1) and k is big enough then the process
{^([znjfc^O < x < 1} will be close to the process {xS(nk);0 < x < 1}. It is
really so and it is a trivial consequence of
STRASSEN'S THEOREM 1 A964). The sequence
sn{x) = bn{S[nx] +(x- I^H X[nx]+1) @ < x < 1; n = 1,2,...)
is relatively compact in C@, l) with probability 1 and the set of its limit points is
S (see notations to Strassen type theorems).The meaning of this statement is that there exists an event Qo C Q of prob-
probability zero with the following two properties:
Property 1. For any u> / Qo and any sequence of integers 0 < n^ < n2 < ...
there exist a random subsequence nk. = rik^u)) and a function / G S such that
snk,(x,uj) —> f(x) uniformly in x € [0,1].
Property 2. For any / G S and u ? Qo there exists a sequence of integersft* = rik(u>,f) such that
snk(x,u) —> f(x) uniformly in x G [0,1].
The Invariance Principle 1 of Section 6.3 implies that the above theorem is equiv-equivalent to
STRASSEN'S THEOREM 2 A964). The sequence {wn{x);0 < x < 1} is
relatively compact in C@,1) with probability 1 and the set of its limit points is Swhere
wn{x)=bnW{nx) (n = l,2,...).
Remark 1. Since |/A)| < 1 for any function / G S and f(x) = x G S, Strassen's
theorem 1 implies Khinchine's LIL.
NON-ACTIVATEDVERSIONwww.avs4you.com
STRASSEN TYPE THEOREMS 81
Consequence 1. For any e > 0 and for almost all u> ? Q there exists a To =
TQ(?,u>) such that if
W{T) > A - e){b{T)Yl for some T > To
then
sup <
Consequence 1 tells us that if W{t) "wants" to be as big in point T as it can
be at all then it has to increase in @, T) nearly linearly (that is to say it has to
minimize the used energy).The proof of Strassen's theorem 2 will be based on the following three lemmas.
LEMMA 8.1 Let d be a positive integer and a\, a^-, • • ¦ ,a-d be a sequence of real
numbers for which
1=1
Further, let
W*{n) = aiW{n) + a2{W{2n) - W{n)) + ¦¦¦ + ad{W{dn) - W{{d - l)n)).
Then
limsup6nW*(n) = 1 a.s. (8.6)n—»oo
and
liminf6nW*(n) = -1 a.s. (8.7)
Proof of this lemma is essentially the same as that of the Khinchine's LIL. The
details will be omitted.
The next lemma gives a characterization of S.
LEMMA 8.2 (Riesz - Sz.-Nagy, 1955, p.75). Let f be a real valued function on
[0,1]. The following two conditions are equivalent:
(i) / is absolutely continuous and Jo{f'Jdx < 1,
and f is continuous on [0, l].
NON-ACTIVATEDVERSIONwww.avs4you.com
82 CHAPTER 8
In order to formulate our next lemma we introduce some notations. For any
real valued function / € C@,1) and positive integer d, let f(d) be the linear
interpolation of / over the points i/d, that is
Sd = {/(d) : / e S}
where Sd C S by Lemma 8.2.
LEMMA 8.3 The sequence {w^(x);0 < x < 1} is relatively compact in Cdwith probability 1 and the set of its limit points is Sd-
Proof. By Khinchine's LIL and continuity of Wiener process our statement
holds when d = 1. We prove it for d = 2. For larger d the proof is similar
and immediate. Let Vn = (W(n),WBn) - W(n))(n = 1,2,...) and a,0 be real
numbers such that a2 + j32 = 1. Then by Lemma 8.1 and continuity of W the
set of limit points of the sequence
y/2n log log n jJ n—1
_
I aW(n) + 0(WBn) -W(n)))°°\ y/2n log log n J n=1
is the interval [—l,+l]. This implies that the set of limit points of the sequence
{bnVn} is a subset of the unit disc and the boundary of the unit circle belongs to
this limit set.
Now let V* = (W(n),WBn) - W(n),W{3n) - W{2n)). In the same way as
above one can prove that the set of limit points of {bnV^} is a subset of the unit
ball of J?3 which contains the boundary of the unit sphere. This fact in itself
already implies that the set of limit points of {6nVn} is the unit disc of R2 and
this, in turn, is equivalent to our statement.
Proof of Strassen's Theorem 2. For each u^fiwe have
sup0<
w (x) - w^(x)\ < sup sup \wn(x + s) - wn(x)\,0<z<l
hence by Theorem 7.13 (cf. also Example 3 of Section 7.2)
limsup wn(x) — wlf'(x)\ = d~1'2 a.s.
NON-ACTIVATEDVERSIONwww.avs4you.com
STRASSEN TYPE THEOREMS 83
Consequently we have the Theorem by Lemmas 8.2 and 8.3 where we also use
the fact that Lemma 8.2 guarantees that 5 is closed.
The discreteness of n is inessential in this Theorem. In fact if we define
wt[x)=btW{tx) (i6[0,l],t>0)
then we have
STRASSEN'S THEOREM 3 A964). The net wt(x) is relatively compact in
C@,1) with probability 1 and the set of its limit points is S.
As an application of Strassen's theorem we sketch the proof of Theorem 7.18
in the special case at = at.
At first we mention that Strassen's theorem implies that
\ims\ipbtJi(t,at) = a1^2 a.s.
t—oo
which can be obtained by considering the function
,( \ - I xa~1/2 if °<x<<*,
M*J-jai/2 if a<x<x
in Strassen's class S (cf. also Theorem 7.13 and Example 3 of Section 7.2).The fact that bt is the right normalizing factor for the liminf also follows from
Strassen's theorem. Let
Ca = — liminf btJ\{t, at).t—*oo
In case a = 1 it is well known that C\ = 1 and this can be obtained by consideringthe function f(s) = -s @ < s < 1) in S. Considering the function f(s) = -s
it is also immediate that Ca > a. Theorem 7.18 claims, however, that equalityholds (i.e.Ca = a) if and only if \/a is an integer; in other cases Ca > a.
Now we show that the Strassen's theorem implies that
liminf 6(t)Ji(t, erf) < -Ca = - (B?" t^"
l) . (8.8)*-»oo y r{r + 1) J
Define the function x(s) as follows: if I/a = r (an integer), then let x{s) = -s.
If I/a = r + t, where r is an integer and 0 < x < 1, then split the interval [0,1]into 2r + 1 parts with the points
u2, = ia i = 0,1,2,..., r
= (i + r)a i = 0,1,2, ...,r.
NON-ACTIVATEDVERSIONwww.avs4you.com
84 CHAPTER 8
Let x(s) be a continuous piecewise linear function starting from 0 (i.e. x[0) = 0)and having slopes
lr(r+l)V/2 .,if u2t < s < u2t+1,r + 1 \ar + 1-t
ar + 1 — tU2i.
It is easily seen that x(s) so defined is in Strassen's class, i.e. x@) = 0, x(s) is
absolutely continuous for 0 < 5 < 1 and Jq x'2(s)ds = 1. Since
x(s + a) - x(s) = -Ca, 0 < s < 1 - a
we have (8.8). Unfortunately we cannot accomplish the proof of Theorem 7.18
by showing that x(s) defined above is extremal within S. In Csaki - Revesz
A979) the proof was completed by some direct probabilistical ideas. The details
are omitted here.
Here we mention a few further applications of Strassen's theorem given byStrassen A964). At first we present the following:
Consequence of Strassen's Theorems 1 and 2. If (p is a continuous func-
functional from C@,1) to R1 then with probability 1 the sequences <p(wn{i)) and
<p(sn(t)) are relatively compact and the sets of limit points coincide with <p[S).Consequently
\ims\ip<p(wn(t)) = \imsuip<p(sn(t)) = sup^>(x) a.s.
n—»oo n-*oo zGS
Applying this corollary to the functional
<p(x) = jT1 x(t)f(t)dt (*€C@,l))
where f(t) @ < t < 1) is a Riemann integrable function with
we obtainr1 r1
limsup / wn(t)f(t)dt = limsup / sn(t)f(t)dtn—*oo JO n—kx> JO
1n f i\
= limsup-V/ - b(n)Si = sup<p(x), (8.9)
NON-ACTIVATEDVERSIONwww.avs4you.com
STRASSEN TYPE THEOREMS 85
and by integration by part we get
Ofi \ 1/2
(F(t))*dt) . (8.10)o /
The above consequence also implies:For.any a > 1 we have
\imsup n-1 (b(n)) J ^l^V\Jo [I — i
in particular
limsupn" 6(n) z-< 1^*1 =3'
a.s.,
n
7t- = 4tt~ a.s.
Remark 2. In order to prove (8.11) we have to prove only that
VVJo A-
This can be done by an elementary but hard calculation.
Similarly we obtain
n)J^—= 2p a.s.
where p is the largest solution of the equation
A - pj^sin (p-^l - pI/2) + cos (A - Pyl*p-') = 0.
A further application given by Strassen is the following. Let 0 < c < 1 and
fi if Si>c(b(i))-\'l
\0 otherwise.
NON-ACTIVATEDVERSIONwww.avs4you.com
86 CHAPTER 8
Consider the relative frequency qn = n'1 ?"_3 ct. We have
Iimsup7n = 1 -
exp (-4(—- 1)) a.s.
Strassen also notes:
"For c = 1/2 as an example we get the somewhat surprising result that with
probability 1 for infinitely many n the percentage of times i < n when 5, >
l/2Bi'loglogi'I/2 exceeds 99.999 but only for finitely many n exceeds 99.9999."
Finally we mention a very trivial consequence of Strassen's theorem.
THEOREM 8.2 The set
{btm+(xt);0<x<l} [t -> oo)
and the sequence
;0 < x < 1J (n - oo)
are relatively compact in C@, l) with probability 1 and the set of their limit pointsis the set of the nondecreasing elements of S. The analogous statements for m(t)and M(n) are also valid.
8.2 Strassen theorems for increments
As we have already mentioned, Khinchine's LIL is a simple consequence of
Strassen's theorem 1. Here we are interested in getting such a Strassen typegeneralization of Theorem 7.13. At first we mention a trivial consequence of
Theorem 7.13.
Consequence 1. For almost all lj G Q and for all e > 0 there exists a To =
T0(e,u) such that for all T > To there is a corresponding 0 < t = t(u,e,T) <
T —
ar such that
W{t + aT) - W{t) > A - e)G(r,aT))-1 « A - e){2aT logTa^I'2 (8.12)
provided that a? satisfies conditions (i), (ii), (iii) of Theorem 7.13.
Knowing Consequence 1 of Section 8.1 we might pose the following question:does inequality (8.12) imply that W(x) is increasing nearly linearly in (t,t +
a-r)l The answer to this question is positive in the same sense as in the case of
Consequence 1 of Section 8.1.
In order to formulate our more general result introduce the following nota-
notations:
NON-ACTIVATEDVERSIONwww.avs4you.com
STRASSEN TYPE THEOREMS 87
(i) I\T(x) = 7(r,ar)(W(t + xaT) - W(t)) @ < x < 1),
(ii) for all u G Cl define the set VT = Vt(uj) C C@,1) as follows:
Vr = (I\r(z) : 0 < t < T - aT},
(iii) for any A C C@,1) and e > 0 denote ?/(A, e) be the ^-neighbourhood of
A in C@,1) metrics, that is a continuous function a(x) is an element of
U(A,e) if there exists an a(x) G A such that supo^^ | a(x) - a(x) \< e.
Now we present
THEOREM 8.3 (Revesz, 1979). For almost all u ? Q and for all e > 0 there
exists a TQ = To(u,e) such that
U{VT{u),e)DS (8.13)
and
U(S,e)DVT(uj) (8.14)
ifT>T0 provided that ax satisfies conditions (i), (ii), (iii) of Theorem 7.13.
To grasp the meaning of this Theorem let us mention that it says that:
(a) for all T big enough and for all s(x) € S there exists a.0<t<T —
aT such
that TtiT(x)@ < x < 1) will approximate the given s(x),
(b) for all T big enough and for every 0 < t < T -
aT the function I\r(a:)@ <
x < 1) can be approximated by a suitable element s(x) G S.
We have to emphasize that in Theorem 8.3 we assumed all the conditions (i),(ii), (iii) of Theorem 7.13. If we only assume conditions (i) and (ii) then we geta weaker result which contains Strassen's theorem 3 in case at = T.
THEOREM 8.4 (Revesz, 1979). Assume that aT satisfies conditions (i) and
(ii) of Theorem 7.13. Then for almost all u G Q and for all e > 0 there exists a
Tq = T0(e,lj) such that
VT{uj)cU{S,e)
if T > Tq. Further, for any s = s(x) G S, e > 0 and for almost all u> G Q there
exist a T = T(e, w, s) and a 0 < t = t(e, u>, s) < T —
a? such that
sup \VttT{x) -s{x)\ <e.0<z<l
NON-ACTIVATEDVERSIONwww.avs4you.com
88 CHAPTER 8
Remark 1. The important difference between Theorems 8.3 and 8.4 is the fact
that in Theorem 8.3 we stated that for every T big enough and for every s(x) G S
there exists a 0 < t < T —
ax such that Tt>T(x) approximates the given s(x); while
in Theorem 8.4 we only stated that for every s(x) G S there exists a T (in fact
there exist infinitely many T but not all T are suitable as in Theorem 8.3) and
a.0<t<T— ax such that Ttj(x) approximates the given s(x).In other words if ar is small (condition (iii) holds true), then for every T (big
enough) the random functions Ytj{x) will approximate every element of S as t
runs over the interval [0, T — ay]. However, if ar is large then for any fixed T the
random functions I\x(a:)@ < t < T — ar) will approximate some elements of S
but not all of them; all of them will be approximated when T is also allowed to
vary.
8.3 The rate of convergence in Strassen's
theorems
Property 1. Strassen's theorems 1 and 2 can be reformulated as follows: forany e > 0 and for almost all uj G H there exists an integer no = no(e,u) > 0 such
that
sn(x,uj) G U(S,e) and wn(x,lj) G U(S,e)
if n > no, equivalently there exists a sequence en = en \ 0 such that
sn(x,uj) G U(S,en) and wn(x,uj) G U(S,en) a.s.
for all but finitely many n.
It is natural to ask how can we characterize the possible en sequences in
the above statement. This question was proposed and firstly investigated byBolthausen A978). A better result was given by Grill A987/A), who proved
THEOREM 8.5 Let
^Mn) = (log log n)"*.
Then
sn(x) G U(S,tl)s[n)) and wn(x) G C/(S,^(n)) a.s.
for all but finitely many n if 6 < 2/3; while for 6 > 2/3
sn{x) ? U(S',^«(n)) and ^n(^) ^ U(S,tl>s{n)) i-o. a.s.
NON-ACTIVATEDVERSIONwww.avs4you.com
STRASSEN TYPE THEOREMS 89
Clearly Theorem 8.5 implies Property 1 of Section 8.1 but it does not contain
Property 2 of Section 8.1. As far as Property 2 of Section 8.1 is concerned one
can ask the following question.Let f(x) be an arbitrary element of S. We know that for all e > 0 and for
almost all u> € Q there exists an integer n = n(e,u) resp. n = n(e, u) such that
sup \sn(x) - f[x)\ < e resp. sup \iVn{x) - f(x)\ < e.
0<z<l 0<z<l
Replacing e by en in the above inequalities, they remain true if en | 0 slowlyenough. We ask how such an en can be chosen. This question was raised and
studied by Csaki. He proved
THEOREM 8.6 (Csaki, 1980). For any f(x) € S and c > 0 we have
sup \wn(x) - f(x)\ < c(loglogn)�/2 i.o. a.s.
0<x<\
and
sup \wn{x) - f{x)\ > -A - c)(loglogn) a.s.
o<z<i 4
for all but finitely many n.
If Jq (f (x)Jdx = a < 1 then a stronger result can be obtained:
THEOREM 8.7 (Csaki, 1980, de Acosta, 1983). If f{x) <E S,Jo{f'{x)Jdx =
a < 1 and c > 0 then
sup \wn(x) - f(x)\ < —
rrj-—r— i.o. a.s.
o<x<i 4A - aI'2 log log n
and
for all but finitely many n.
In case fo[f'(x)Jdx = 1 the best possible rate is available only for piecewise linear
functions. Let f(x) be a continuous piecewise linear function with /@) =0 and
f'(x) = fa if a,_i < x < a, (t = 1,2,... fc)
where a0 = 0 < ax < ... < a*-! < a* = 1. Then we have
NON-ACTIVATEDVERSIONwww.avs4you.com
90 CHAPTER 8
THEOREM 8.8 (Csaki, 1980). If f[x) is defined as above and f*(f'(x)Jdx =
1 then for any s > 0
sup \wn{x) - /(x)| < ^IZ2-SIZB~1IZ{1 + e)(loglogn)�/3 i.o. a.s.
0<z<l
and
sup \wn{x) - f{x)\ > ^lz2-hlzB-l'z{l - e)(loglogn)�/3 a.s.
0<z<l
for all but finitely many n where
Remark 1. Theorems 8.5 resp. 8.8 imply that for any /? < 2/3
W[t) < («Bloglog« + (loglog«I^)I/2 a.s., (8.15)
if t is big enough resp.
W{t) > (t Bloglog« - A + ^(loglogO^V/^1/3^-1/3)) i.o. a.s. (8.16)
(8.15) and (8.16) clearly imply the Khinchine's LIL but they are much weaker
than EFKP LIL of Section 5.2 (cf. Consequence 1 of Section 5.2).
Remark 2. Applying Theorem 8.8 for f(x) = 0 @ < x < 1) we obtain (8.16) as
a special case.
Remark 3. It looks an interesting question to find a common generalization of
the results of Section 8.2 and those of Section 8.3, i.e. to investigate the rate of
convergence in Theorems 8.3 and 8.4. This question was studied by Goodman
and Kuelbs A988).
Remark 4. Csaki (personal communication) recently found the generalizationof Theorem 8.8 in the case when f(x) is a quadratic function (and of course
1
8.4 A theorem of Wichura
We have seen that Strassen's theorem is a natural generalization of Khinchine's
LIL. Wichura proposed to find a similar (Strassen type) generalization of the
Other LIL (cf. E.9)). He proved
NON-ACTIVATEDVERSIONwww.avs4you.com
STRASSEN TYPE THEOREMS 91
WICHURA'S THEOREM (Wichura, 1977, Mueller, 1983). Consider the
sequence
yp|M ) H||; 0< u<
J \ n )
(n = 1,2,...) and the net
tit(u) = | 1logl�" 1 sup \W(tx)\; 0 < u < 1 }> (t -> oo).
^V l J *<« J
Let ? 6c t/ic set o/ nondecreasing, nonnegative functions g on [0, l] satisfying
i:Then with probability 1, the set of limit points of sn(u) resp. iOj(u) in the weak
topology, as n /* oo resp. t / oo is Q.
Remark 1. In order to see that this theorem implies the Other LIL we onlyhave to prove that g(l) > y| for any g(-) € Q.
NON-ACTIVATEDVERSIONwww.avs4you.com
Chapter 9
Distribution of the local time
9.1 Exact distributions
Let
Px(fc) =min{n:5n = k} {k = 1,2,.. .)•
Then
Hence by Theorem 2.4 we obtain
THEOREM 9.1
(fc = l,2,...,n = l,2,...).
j=°x[~2~yEspecially
Consequently
1) = 2m + 1} = 2-2~-1(m + I)�^ (9.1)
Note that /?iBA: + 1) takes only odd, /?iBA:) takes only even numbers.
93
NON-ACTIVATEDVERSIONwww.avs4you.com
94 CHAPTER 9
THEOREM 9.2 Let p0 = 0 and pk = min{j : j > pk-i, S}¦ = 0}(/c = 1,2,...).Then px, p2
—
p\,pz—
P2, ¦ • • I5 ° sequence of x.i.d.r.v's wxth
¦p/_ — Ot\ — O-2*+1i--1 I 1 ft- 1 O ^ (Q O\^ J \ Ic I /
\ /
and
> 2n) = 2'2n[ n) = P{52n = 0}.
Proof. The statement that Pi,P2~ Pi,Pz~ P2,- ¦ • are i.i.d.r.v.'s taking only even
values is trivial. Hence we prove (9.2) only.
J>{Pl = 2k} = ip{Pl =2k\Xi = +1} + \p{pi = 2k\Xx = -1}.2 &
Clearly
l= 2k | Xx = +1} = PI/?! = 2A: | Xx = -1} = P{pi(l) =2k- 1}. (9.3)
Hence by (9.1) we have (9.2). The second statement of Theorem 9.2 is a
simple consequence of (9.2).
Remark 1. A simple calculation gives
f = 2k) =
Hence the particle returns to the origin with probability 1, i.e. we obtained a
new proof of Polya Recurrence Theorem of Section 3.1. However, observe that
i.e. the expectation of the waiting time of the recurrence is infinite.
Consider a random walk {5*; A: = 0,1,2,...} and observe how many times a
given x(x = 0, ±1,±2,...) was visited during the first n steps, i.e. observe
?{x,n)=#{k:0<k<n,Sk = x}.
The process f (a:, n) (of two variables) is the local time of the random walk {Sk}.
NON-ACTIVATEDVERSIONwww.avs4you.com
DISTRIBUTION OF THE LOCAL TIME 95
THEOREM 9.3 For any k = 0,1,2,..., n; n = 1,2,... we /iave
n) = k}= P{?@,2n + 1) = *} = 2
Equivalently
*-i (on - A> 2n} = P{e(O, 2n) < A:} = 2�" ^ 2'
J.
i=o V n/
Proof. (9.3) implies that the distribution of pi is identical with that of pi[l) +1.
It follows that the distribution of pk— k is identical to that of Pi(k), i.e.
-k>n} = P{Pl{k) >n}= P{Mn+
Further, we have
P{f@,2n) =k} = P{pk < 2n,pk+1 > 2n)
which implies the Theorem.
Applying Theorems 9.1 and 9.3 we can get the distribution of f (a:, n). In fact
we have
THEOREM 9.4 Let x > 0. Then for any k = 1,2,...
,n) = k}= j;*P{e(x,n) = k | Pl(x)j=x
J=X
r
and for k = 0
l=n+l*
\ o '
NON-ACTIVATEDVERSIONwww.avs4you.com
96 CHAPTER 9
Theorem 9.3 gave the distribution of the number of zeros in the path So, Si,...,
S^n+i- We ask also about the distribution of the number of those zeros where
the path changes its sign. Let
0(n) = #{A; : 1 < k < n,Sk.xSk+i < 0}
be the number of crossings (sign changes). Then as a trivial consequence of
Theorem 9.3 we obtain
THEOREM 9.5
P{0Bn + !)=*} = ? P{0Bn + l) = k | ?@,2n) = j}P{?@,2n) = j}j=k
)() = (j?k \k) 2>\ n ) 22n~> 22n \n +
= 2P{52n+1 =2k + 1}.
Proof. Observe that P{5jfe_i5jfe+1 < 0 | 5* = 0} = 1/2.
THEOREM 9.6
P{ max Si > n} = Bn)� (n=l,2,...).
Proof. It is a trivial consequence of Lemma 3.1.
THEOREM 9.7 For any k = ±1, ±2,... and 1 = 0,1,2,... we have
(9.4)
)' (9-5)
= l, E(?(A:,p1)-lJ=4A:-2. (9.6)
Proof. Without loss of generality we may assume that k > 0. Then
{?(*,*) = o} = {Xi = -1} u {Xi = i,s2 # k,s3 # k,...,spi_i # k}.
Hence by Lemma 3.1
1 1 A: — 1 2A;-1
NON-ACTIVATEDVERSIONwww.avs4you.com
DISTRIBUTION OF THE LOCAL TIME 97
and (9.4) is proved.For any x = ±1, ±2,... define
po[x) = 0,
Pl{x) =inf{l:l>0,Sl = x},
pi+1{x) = inf{/ : / > Pi{x), S, = x} (i = 0,1,2,...).
Then in case m > 0 we have
{t[k,Pi) = m} = {0 < Pl(k) < P2{k) <...< pm[k) <Pl
Hence, again by Lemma 3.1
P{€(*,*) = -I =
-2l [E { j ) B) (j- A \~
\2k)
2k
2k
and (9.5) is also proved.(9.6) is a simple consequence of (9.4) and (9.5).Define the r.v.'s ftn as follows: f2n A = 1,2,...) is the number of those terms
of the sequence 5l5 S2,..., ^n which are positive or which are equal to 0 but the
preceding term of which is positive. Then ftn takes on only even numbers and
its distribution is described by
THEOREM 9.8
B*)(^lfJ�B (* = 0,l,.-.,»). (9-7)
Proof. (Renyi, 1970/A). Clearly
{f2n = 0} = {M2+n = 0}.
Hence by Theorem 2.4
n\2~2n,
i.e. (9.7) holds for k = 0. It is also easy to see that (9.7) holds for n = 1, k = 0,1.Now we use induction on n. Suppose that (9.7) is true for n < N—l and consider
NON-ACTIVATEDVERSIONwww.avs4you.com
98 CHAPTER 9
= 2k} for 1 < k < N - 1. If <;2N = 2k and 1 < k < N - 1 then the
sequence SX,S2, ¦ ¦ ¦ ,S2N has to contain both positive and negative terms, and
thus it contains at least one term equal to 0. Let pi = 21. Then either Sn > 0
for n < 21 and S2i = 0 or Sn < 0 for n < 21 and 52j = 0. Both possibilities have
the probability
(cf. (9.2)).Now if Sn > 0 for n < 21 and S2l = 0, further if $2N = 2k, then among the
numbers S2l+i,... ,S2N there are 2k — 21 positive ones or zeros preceded by a
positive term, while in case Sn < 0 for n < 21, S2l = 0 and $2N = 2k, the number
of such terms is 2k. Hence
W = 2/}P{f2*_2, = 2k- 21}
+ \fl p(^i = 2OPU2*-* = 2A;}1
1=1
and we obtain (9.7) by an elementary calculation.
It is worthwhile to mention that the distribution of the location of the last
zero up to 2n, i.e. the distribution of
*Bn) = max{Jk : 0 < k < n, S2k = 0}
agrees with the distribution of $2n. In fact we have
THEOREM 9.9
Proof. Clearly by Theorem 9.2
P{#Bn) = 2k} = P{S2k = 0}P{Pl > 2n - 2k} = P{52Jfe = 0}P{52n_2Jfe = 0}.
Hence the Theorem.
The distribution of the location of the maximum also agrees with those of fn
and \&(n). In fact we have
THEOREM 9.10 Let
fi+{n) = inf{k:0<k<n for which S{k) = M+(n)}.
NON-ACTIVATEDVERSIONwww.avs4you.com
DISTRIBUTION OF THE LOCAL TIME 99
Then
P{M+Bn)=A;} =
(B[k/2]\Bn-2[k/2\\[m){n-[k/2]J»-2n A; = 0.
n
Proof. Clearly the number of paths for which
is equal to
Hence
P{M+Bn)
the number of
} = P{S0 -
= P{50 -
paths
{Si >
<Sk,l
for which
0,52 >0,
?! < Sk,..
h<sk,..
Sk, Sk+1 < Sk,..., S2n < Sk}
?! > 0, S2 > 0,..., Sk > 0}'P{S1 > 0,..., 52n_jfe > 0}.
Then we obtain Theorem 9.10 by Theorem 2.4.
9.2 Limit distributions
Applying the above given exact distributions and the Stirling formula we obtain
THEOREM 9.11
^ / y« *\ 1 1*1.
v e av, [\).a)*-
(9.9)
where \&k\ < 1,
(9.10)
NON-ACTIVATEDVERSIONwww.avs4you.com
1Oo CHAPTER 9
HmPJn-^eCO.n) < x\ = lim P {n-l<2t[z,n) < x}n—>oo I¦» \ / j n—>oo «. '
e'u2/2du (z = ±l,±2,...), (9.11)
{ n J n~>0° I n J r»->0° I n
2= -arcsinV* @ < x < 1). (9.12)
7T
Remark 1. (9.12) is called arcsine law. It is worthwhile to mention that by(9.12) we obtain
limp(o,45< - < 0,55} = 0,063769...n—00 ^ n )
and
lim P (^ < 0, l} = lim P (^ > 0,9} = 0,204833...
The exact distribution (9.7) of ftn als° implies that the most improbable value
of ftn is n and the most probable values are 0 and 2n. In other words with a bigprobability the particle spends a long time on the left-hand side of the line and
only a short time on the right-hand side or conversely but it is very unlikely that
it spends the same (or nearly the same) time on the positive and on the negativeside.
9.3 Definition and distribution of the
local time of a Wiener process
It is easy to see that the number of the time points before any given T, where a
Wiener process W is equal to a given x, is 0 or 00 a.s., i.e. for any T > 0 and
any real x
#{*:0< t <T,W(t) =x) = 0oroo a.s.
Hence if we want to characterize the amount of time till T which the Wiener
process spends in x (or nearby) then we have to find a more adequate definition
than the number of time points. P. Levy proposed the following idea.
Let H(A,t) be the occupation time of the Borel set A C R1 by W(-) in the
interval @,t), formally
H{A,t) = \{s:s<t,W{s)
NON-ACTIVATEDVERSIONwww.avs4you.com
DISTRIBUTION OF THE LOCAL TIME 101
where A is the Lebesgue measure.
For any fixed t > 0 and for almost all u; € Cl the occupation time H(A,t)is a measure on the Borel sets of the real line. Trotter A958) proved that this
measure is absolutely continuous with respect to the Lebesgue measure and its
Radon - Nikodym derivate rj(x,t) is continuous in (x,t). The stochastic process
ri(x,t) is called the local time of W. (It characterizes the amount of time that
the Wiener process W spends till t is "near" to the point x.) Our first aim is to
evaluate the distribution of the r.v. 77@, t).In fact we prove
THEOREM 9.12 For any x > 0 and t > 0
9 rz
(9.13)1
7T Jo
(N) (N)Proof. For any N = 1,2,... define the sequence 0 < T\ = t{
'< r2 = r2v
;< ...
as follows:
Tx = M{t:t>0,\W[t)\ = N-1},r2 = inf{* :t>ru \W(t) - W{Tl)\ = N-1},
ri+1 = inf{* : * > r.-, \W{t) - W{n)\ = N'1},
(cf. Skorohod embedding scheme, Section 6.3) and let
siN)=W{rk) (A; = 1,2,...),
u = u^N) = max{i : r, < 1}.
Note that Ti,t2—
Ti,... is a sequence of i.i.d.r.v.'s with
Et"! = N~2 and ErJ < 00 (9.14)
(cf. F.1) and Theorem 6.3).The interval (rj,rl+i) will be called type (a,6) (a = jiV�,^— a\ = N-1,j =
0,1,2,...) if \W(Ti)\ = a and |W(ri+i)| = 6. The infinite random set of those
j's for which (r,-,Tj+1) is an interval of type (a, 6) will be denoted by I^(a,b) =
I(a, 6). It is clear that |W(*)| can be smaller than JV� if t is an element of an
interval of type (O,^�), {N-\0) or (iV�^^�). Let
A = AW = {i:0<i<u,ie /(O,^�) U
NON-ACTIVATEDVERSIONwww.avs4you.com
102 CHAPTER 9
Then by the law of large numbers and (9.14)
7Ti) -»1 a.s. (JV-oo). (9.15)
(In fact (9.15) can be obtained using the "Method of high moments" of Section
4.2; to obtain it by "Gap method" seems to be hard.)Studying the local time of W(-) in intervals of type (N~l,2N~1), we obtain
5>(O,rl+1)-r7(O,r,)) = O a.s. (9.16)
for any N = 1,2,... where
B = BN = {i : 0 < i < u,i € I{N~X,2N~1)}. (9.17)
((9.16) follows from the simple fact that for almost all w € fl there exists an
e0 = eo{u,N) such that | W(t) |> e0 if t e U.esK^.+i)-)Hence
1,@,1) = Jim ^ ?(rI+1 - r.-) a.s. (9.18)N^°° 2
i€A
Then, taking into account that limjv_oo N~2uN = 1 a.s., (9.11), (9.15) and
(9.18) combined imply that
Jl[X (9.19)
and Theorem 9.12 follows from (9.19) and from the simple transformation: for
any T > 0
{r,[x,tT),x e R\0 <t<l} = {Tll2r)[xT-ll2,t),x eR\0<t< l}. (9.20)
Theorem 9.12 clearly implies that for any x € R1 we have
Levy A948) also proved
THEOREM 9.13 For any xe R1, T > 0 and u > 0 we have
NON-ACTIVATEDVERSIONwww.avs4you.com
DISTRIBUTION OF THE LOCAL TIME 103
To evaluate the distribution of 77 (t) = s\ip_O0<x<O0rj(x,t) is much harder. This
was done by Csaki and Foldes A986). They proved
THEOREM 9.14
where 0 < j\ < j2 < ... are the positive zeros (roots) of the Bessel function
Jo(x) = Io(ix) and for any k = 1,2,...
4ak =
. 2 • »
sm j*
6, = 4 I-] ^
Jk7rJ0(A;7r) \J0{kn)) ]'
Ji(A;7r)
1
i
Furthermore
i / 2j,2\<z^«aiexp f as z -¦ 0.
Remark 1. The proof of Theorem 9.14 is based on a result of Borodin A982),who evaluated the Laplace transform of the distribution of t~ll2r){i).
NON-ACTIVATEDVERSIONwww.avs4you.com
Chapter 10
Local time and Invariance Principle
10.1 An Invariance PrincipleThe main result of this Section claims that the local time f(:r,n) of a random
walk can be approximated by the local time ri(x, n) of a Wiener process uniformlyin x as n —¦ oo. In fact we have
THEOREM 10.1 (Revesz, 1981). Let {W(t),t > 0} be a Wiener process de-
defined on a probability space {fl, J,P}. Then on the same probability space Q one
can define a sequence Xi, X2,... of i.i.d.r.v. 's with P(X, = 1) = P(X, = —1) =
1/2 such that
lim rT1/4~esup|f(:r,n) - r)(x,n)\ =0 a.s. A0.1)n—>oo
x
for any e > 0 where the sup is taken over all integers, r\ is the local time of W
and ? is the local time of Sn = Xi + X2 H h Xn.
For the sake of simplicity, instead of A0.1) we prove only
lim n-<~e|f@,n) - ri@,n)\ =0 a.s. A0.2)n—>oo
for any e > 0. The proof of A0.1) does not require any new idea. Only a more
tiresome calculation is needed.
Proof of A0.2). Define the r.v.'s r0 = 0 < n < t2 < ... just like in Section
6.3. Further let 1 < fix < \i2 < ... be the time-points where the random walk
{Sk} = {W(rk)} visits 0, i.e. let
Mi = min{Jk : k > 0, W{rk) = Sk = 0},M2 = min{A;: k > fiuW[Tk) = Sk = 0},
105
NON-ACTIVATEDVERSIONwww.avs4you.com
106 CHAPTER 10
Mn = min{A; : k > Hn-i,W{rk) = Sk = 0},
Then
f@,n) = max{A; : \ik < n}
and€@.*)
The proof of Theorem 9.12 implies
Hence
() a.s. (k - oo).
F.1) easily implies that
r* = k + o (it'+e) a.s.
Then A0.2) easily follows from
f(O,Jt) =o(k*+e) a.s. (ife-»oo) A0.3)
and
sup {p @,j + k^') - v{O,J)) = o (k*+°) a.s. {k -> oo). A0.4)
A0.3) and A0.4) can be easily proved. Their proofs are omitted here because
more general results will be given in Chapter 11.
Remark 1. It turns out that the rate of convergence in Theorem 10.1 is nearlythe best possible. In fact Remark 3 of Section 11.5 implies that if a Wiener
process W(-) and a random walk {Sn} are defined on the same probability space
then
limsupn~1/'4sup|^(x,n) - ?7(x,n)| > 0 a.s. A0.5)n—>oo x
However, the answer to the following question is unknown. Assume that a Wiener
process and a random walk are defined on the same probability space and
lim n~a I f@, n) -
r\ @, n) \ = 0 a.s.n—*oo
What can be said about a?
NON-ACTIVATEDVERSIONwww.avs4you.com
LOCAL TIME AND INVARIANCE PRINCIPLE 107
Remark 2. It can be also proved that in Theorem 10.1 the random walk Sn and
the Wiener process W(t) can be constructed so that besides A0.1)
\Sn-W{n)\ =O(logn) a.s.
Remark 3. A trivial consequence of Theorem 10.1 is
Km'P{n-1t2?{n) < z) =P{r?(l) < z) A0.6)
for any z > 0 where f(n) = maxx ?(x,n) (cf. Theorem 9.14).
10.2 A theorem of Levy
Theorem 10.1 tells us that the properties of the process ?(x,n) are the same
(or more or less the same) as those of rj(x,n). In other words studying the
behaviour of one of the processes ?(x,n),ri(x,n) we can automatically claim
that the behaviour of the other process is the same. The main results of the
present section tell us that the properties of f@, n) resp. rj@,n) are the same
as those of M+(n) resp. m+(n). Hence the theorems proved for M+(n) resp.
m+(n) will be inherited by f@, n) resp. rj@,n).Let
y{t) =m+{t)-W{t) (t>0)and
Y[n) =M+{n)-S{n) (n = 0,1,2,...).Then a celebrated result of P. Levy reads as follows (see for example, KnightA981), Theorem 5.3.7):
THEOREM 10.2 We have
{y{t),m+{t);t > 0}l{\W{t)\,r,{0,t);t > 0},
i.e. the finite dimensional distributions of the vector valued process {y(t), m+(t);t > 0} are equal to the corresponding distributions of {\W(t)\,ri(Q,t);t > 0}.
In order to see the importance of this theorem, we mention that applyingthe LIL of Khinchine (Section 4.4, see also Theorem 6.2) for m+(t) as a trivial
consequence of Theorem 10.2 we obtain
Consequence 1.
limsup /^ =1 a.s. A0.7)t-oo B* log log*I/2
NON-ACTIVATEDVERSIONwww.avs4you.com
108 CHAPTER 10
In fact the Levy classes can be also obtained for rj(Q,t).Applying Theorem 10.1, Consequence 1 in turn implies
Consequence 2.
lfi =1 a.s. A0.8)msupplfivn—oo y/2n log log fl
Remark 1. A0.7) was proved (directly) by Kesten A965). A0.8) is due to
Chung and Hunt A949).A natural question arises: what is the analogue of Theorem 10.2 in the case
of a random walk? In fact we ask: does Theorem 10.2 remain true if we re-
replace W(t),y(t),m+{t),rl{0,t) by S{n),Y{n),M+{n) and f@,n) respectively?The answer to this question is negative, which can be seen by comparing the
distributions of f@,2n) and M+[2n) (cf. Theorems 2.4 and 9.3).In spite of this disappointing fact we prove that Theorem 10.2 is "nearly true"
for random walks. In fact we have
THEOREM 10.3 (Csaki - Revesz, 1983). Let Xi,X2,... be a sequence ofi.i.d.r.v.'s with P(Xi = 1) = P(Xi = —1) = 1/2 defined on a probability space
{fi, J,P}. Then one can define a sequence Xi, X2,... of i.i.d.r.v. 's on the same
probability space {Q, J,P} such that P(XX = 1) = P(XX = -1) = 1/2 and forany e > 0
n-'\Y{n) -\S[n)\ | -* 0 a.s.
and
n-1/4"e|M+(n) -?@,n)| -* 0 a.s.,
where
M+(n) = max S(A:), 5@) = 0, S{n) = Y^Xk (n = 1,2,...),°-*-n
*=i
Y{n) =M+{n) - S[n).
Remark 2. This theorem is a bit stronger than that of Csaki - Revesz A983).The proof is presented below.
Remark 3. Consequence 2 can also be obtained by applying the LIL of Khin-
chine (cf. Section 4.4) for M+ and Theorem 10.3.
Theorem 10.3 tells us that the vector (|5(n)|, f@,n)) can be approximatedby the vector (Y(n),M+(n)) in order n1/4�"*. Unfortunately we do not know
what the best possible rate here is. However, we can show that by considering
NON-ACTIVATEDVERSIONwww.avs4you.com
LOCAL TIME AND INVARIANCE PRINCIPLE 109
the number of crossings €>(n) instead of the number of roots f@, n), better rates
can be achieved than that of Theorem 10.3. Let
e(n) = #{Jt : 1 < k < n, S{k - l)S(k + 1) < 0}
be the number of crossings. Then we have
THEOREM 10.4 (Csaki - Revesz, 1983 and Simons, 1983). Let XUX2,...be a sequence of i.i.d.r.v.'s with P(Xi = 1) = P(Xi = -l) = 1/2 definedon a probability space {fl, J,P}. Then one can define a sequence Xi,X2,...of i.i.d.r.v.'s on the same probability space {fi, J,P} such that P(Xi = 1) =
P(Xi = -1) = 1/2 and
|M+(n)-20(n)| < 1, A0.9)
|F(n)-|5(n)||<2, A0.10)
for any n = 1,2,... where
M+{n) = max 5 (Jfc), 5@) = 0, S(n) = J^ Xk (n = l,2,...),
Y{n) =M+{n)-S{n).
Proof. Let
tx = min{i : i > 0, S{i - lM(i + 1) < 0},t2 = min{i : i > r,, S{i - l)S{i + 1) < 0},
= min{t : i > n,S{i - l)S{i + 1) < 0},
andif
xxxj+l if
if TJ + 1 < J <
This transformation was given by Csaki and Vincze A961). The following lemma
is clearly true.
NON-ACTIVATEDVERSIONwww.avs4you.com
HO CHAPTER 10
LEMMA 10.1
(i) X\, X2,... is a sequence of i.i.d.r.v. 's with
00
S{k)-
(Hi) 26(r,) =2/ = 5(r«) =M+(r,), / = 1,2,....
(iv) For any n < n < rJ+1 we have &(n) =1,21 < M+(n) < 2/ +1, consequently
0<M+(n)-26(n) < 1.
(v)J 1)| if r, +
if k =
therefore
?{k)=M+{k)-~S{k) < \S{k + l)\ < \S{k)\ + l
and
Y{k) = M+{k) - 5(ik) > \S{k + 1)| - 1 > \S{k)\ - 2.
This proves Theorem 10.4.
Proof of Theorem 10.3. Clearly we have
n-i/4-'|f@,n) - 20(n)| -* 0 a.s. A0.11)
Hence we obtain Theorem 10.3 as a trivial consequence of Theorem 10.4.
Applying the Invariance Principle 1 (cf. Section 6.3), Theorems 10.2 and 10.4
as well as A0.11) we easily obtain
Consequence 3. (Csaki - Revesz, 1983). On a rich enough probability space
{fl, J,P} one can define a Wiener process {W(t);t > 0} and a sequence Xu X2,...
of i.i.d.r.v.'s with PpG = 1) = T{X1 = -1) = 1/2 such that
\S{n)-W{n)\ =O(logn) a.s.,
|2e(n)-»7(O,n)| =O(logn) a.s. A0.13)
NON-ACTIVATEDVERSIONwww.avs4you.com
LOCAL TIME AND INVARIANCE PRINCIPLE 111
and for any e > 0
|e(O,n)-17@,11I = o(n1/*+«). A0.14)
Remark 4. Hence we obtain a new proof of Theorem 10.1 when only a fixed x
is considered.
Remark 5. Having Theorem 10.3, Theorem 10.2 can be easily deduced (cf.Csaki - Revesz 1983, Simons 1983).
Question. Is it possible to define two random walks {S^} and {S^} on a
probability space such that
rra|fW@,n)-2eB)(n)| -* 0 a.s.
for some 0 < a < 1/4 (cf. A0.14)) where fA)@,n) is the local time of S^ and
€>B) is the number of crossings of 5^? If a positive answer can be obtained,then in A0.11) a better rate can be also obtained. However, if the answer to
this question is negative, then A0.11) also gives the best possible rate (exceptthat perhaps ne can be replaced by some logn power). Hence this question is
equivalent to the question of Remark 1 of Section 10.1.
Now we formulate another trivial consequence of Theorem 10.2.
Consequence 4. Let
7{T)= max (W[u)-W[v))V ' 0<u<v<TX
be the maximal fall of W(-) in [0,T|. Then
'2ir n
n *€ LLC(J(T)),
\ 8 log logn/
P {T-X'2J{T) <x}= G{x) = H{x)where G(-) and #(•) are defined in Theorem 2.13.
Proof. Observe that J{T) = maxo<t<r y[t) and apply Theorems 10.2, 2.13, the
LIL of Khinchine (Section 4.4) and the Other LIL E.9).It is also interesting to study the properties of J{T) for those T"s for which
W (T) is very big. In fact we prove
NON-ACTIVATEDVERSIONwww.avs4you.com
112 CHAPTER 10
THEOREM 10.5 Let C\ and C2 be two positive constants. Then there exists
a sequence 0 < tx = ?i(ur, Ci, C2) < t2 = ^(^J Ci> C2) < ... such that
' '
\loglog*n
if
1/2
and W{tn) > CMtn))'
7T2
The proof of the above theorem is based on the following:
THEOREM 10.6 (Mogul'skii, 1979).
limuMogP ( sup \W{t)-tW{T)\ < uTl'2\ = ~.
"->o ^o<t<r J 8
Proof of Theorem 10.5. Observe that the conditional distribution of
{W{t), 0<t<T} given W[T) = CF(r))�
is equal to the distribution of
{BT{t) + Ct{Tb{T)Y\ 0<t<T}.
Further, the maximal fall of BT[t) + Ct{Tb{T))~l is less than or equal to
2maxo<t<r |Br(OI- Hence we obtain
-1
f-exp (-ir^rloglogT) exp("Ci2 loglogT) -
where BT(t) = W(t) - tW[T) @ < t < T) and the proof follows by the usual
way (cf. e.g. the proof of the LIL of Khinchine, Section 4.4).
NON-ACTIVATEDVERSIONwww.avs4you.com
Chapter 11
Strong theorems of the local time
11.1 Strong theorems for ?(z,n) and
The Recurrence Theorem (cf. Section 3.1) clearly implies that for any x =
0,±l,±2,...lim i(x,n) = 00 a.s. (HI)
n—»oov '
In order to get the rate of convergence in A1.1) it is enough to observe that byTheorem 10.3 the limit behaviour of f@, n) (and consequently that of ?(x, n)) is
the same as that of M+. Hence by the EFKP LIL (cf. Section 5.2) and by the
Theorem of Hirsch (cf. Section 5.3) we obtain
THEOREM 11.1 The nondecreasing function
if and only if
The nonincreasing function
if and only if
n=lU
where x is an arbitrary fixed integer.
Having Theorem 10.2 (instead of Theorem 10.3) and Theorem 6.2 or applyingTheorem 11.1 and Theorem 10.1 we obtain
113
NON-ACTIVATEDVERSIONwww.avs4you.com
114 CHAPTER 11
THEOREM 11.2 Theorem 11.1 remains true if we replace ?(•,•) by v(-,-)-
Remark 1. Theorem 11.1 was proved originally by Chung and Hunt A949).Theorem 11.2 is due to Kesten A965).
The study of ?(n) is much harder than that of ?(x,n). However, havingTheorem 9.14 (cf. also A0.6)) one can prove
THEOREM 11.3 (Kesten, 1965, Csaki - Foldes, 1986).
Iimsup6(n)f(n) = lim sup 6(?) 77B) = 1 a.s. (H-2)t—oo
liminfn-1/2(loglognI/2e(n) = liminf r1/2(loglog*I/2r?Mn—»oo t—»oo
= 1 = 21/2ji as- A1.3)
where j\ is the first positive root of the Bessel function Jo(x).
Remark 2. A1.2) is due to Kesten A965). A1.3) is also due to Kesten without
obtaining the exact value of 7.
The result of Csaki and Foldes A986) is much stronger than A1.3). In fact
they proved:
THEOREM 11.4 Let u(t) > 0 be a nonincreasing function such that
oo u(t) = Q,u{t)tll2 is nondecreasing and lim^oo u{i)tll2 = 00. Then
u{i) e LLC (t-lf2Ti(t)) and u{n) € LLC (n�/^(n))if and only if
(Remark 3. The proof of Theorem 11.4 is based on Theorem 9.14.
The upper classes of rj(t) and those of f(n) were also described by Csaki
A989). He proved
THEOREM 11.5 Let a(t) > 0 (t > 1) be a nondecreasing function. Then
a(t) E UUC (r1/2»7(O) and a{n) ? UUC (fTif and only if
a3(t) ( a2f°°a3(t) ( a2(t)\ ,
L -rexp -41 r<o°-
NON-ACTIVATEDVERSIONwww.avs4you.com
STRONG THEOREMS OF THE LOCAL TIME 115
Since ?(Q,pn) = n, i.e. pn is the inverse function of ?@, n), by Theorem 11.1
we can also obtain the Levy classes of pn. Here we present only the simplestconsequence.
THEOREM 11.6 For any e > 0 we have
n2(lognJ+e€UUC(pn),n2(lognJ-e€ULC(pn),
11.2 Increments of rj(x,t)First we give the analogue of Theorem 7.13.
THEOREM 11.7 (Csaki - Csorgo - Foldes - Revesz, 1983). Let at{t > 0) be a
nondecreasing function of t for which
CO 0 < at < t,
(ii) t/at is nondecreasing.
Then
sup (r)(x,s + at) — rf(x,s))t-»oo
= limsup 6t(r)(x,t) — r}{x,t — at)) = 1 a.s.
If we have also
(Hi)
t—»oo
then
lim(log(«/at))(loglog«)l
= oo—»oo
lim 6t sup (r)(x,s + at) — r)(x,s)) = 1 a.s.t—oo o<*<t-a,
for any fixed x € R1 where 6t = a;1/2(log(t/at) + 2 log log t)~ll2.
By Theorem 10.2 as a trivial consequence of the above Theorem we obtain
THEOREM 11.8 Theorem 11.7 remains true replacing r)(x,t) by m+(t).
NON-ACTIVATEDVERSIONwww.avs4you.com
116 CHAPTER 11
Remark 1. Clearly
sup {m+{s + at)-m+{s))< sup {W{s + at) - W{s)). A1.4)
Comparing 6t and it of Theorem 7.13 we obtain in A1.4) that for a sequence
t = tn | oo we may have strict inequality whenever (iii) does not hold true.
The investigation of the largest possible increment in t when x is also varyingseems to be also interesting. We obtained
THEOREM 11.9 (Csaki - Csorgo - Foldes - Revesz, 1983). Let at{t > 0) be
a nondecreasing function oft satisfying conditions (i) and (ii) of Theorem 11.7.
Then
limsup^t sup sup (rf(x,s + at) - ri(x,s)) = 1 a.s.
t-»oo xCR1 0<s<t-at
If we also assume that (iii) of Theorem 11.7 holds then
lim^sup sup [rf(x, s + at) — rj(x,s)) = 1 a.s.'—°° 0<<
To find the analogue of Theorem 7.20 seems to be much more delicate. At
first we ask about the length of the longest zero-free interval. Let
r(t) = sup{a : for which 30 < s < t - a such that r)(Q,s + a) - r)(Q, s) = 0}
be the length of the longest zero-free interval. Then we have
THEOREM 11.10 (Chung - Erdos, 1952). Let f(x) be a nondecreasing func-function for which l\mx-.O0f(x) = oo,x/f(x) is nondecreasing and limz—oo x/f(x) =
oo. Then
' t1" m)€ uuc(r(())
if and only if
-
Remark 2. Originally this theorem was formulated for random walk instead of
Wiener process.
Example 1. Since L{f) < oo if f(x) = (logxJ+e(er > 0) and L{f) = oo if
f(x) = (logxJ, we obtain
( W)s uuc(r(())
NON-ACTIVATEDVERSIONwww.avs4you.com
STRONG THEOREMS OF THE LOCAL TIME 117
and
111 - (ii?)€ ULC<r"»-
or equivalently
lim inf inf G7@,5 + at) - 77@,5)) = lim infG7@, t) - 77@, t- at)) > 0t—+oo O^s^t — &t t—*oo
and
lim inf inf G7@,5 + at) - 77@,5)) = lim infG7@,*) - 77@, t- at)) = 0
This example shows that the study of the lim inf properties of info<s<t-a, (^ @,5 +
at) — 7?@,5)) (i.e. the analogue of Theorem 7.20) is interesting only if at >
t(l - (log*)�). This question was studied by Csaki and Foldes A986). Theyproved
THEOREM 11.11 Let fi(t) = t^-at)'1 be a nondecreasing function for which
t/fi(t) is also nondecreasing and limt—oot/fi{t) = cx>,\imt-.oo fi(t) = 00. Fur-
Further, let f2(t) be a nonincreasing function for which lim^oo f2{t) = O,t1/2f2(t) is
nondecreasing and limt_>Oo*1^2/2@ = °°- Then
(t) e LUC { inf G7 @,5 + at) - 77 @,5))}if
L{h) = ao or V (/,) = 00,
and
t^fiit) e LLC {inf G7@,5 + a,) - 77@,5))}if
L{h) < 00 and L*(f2) < 00,
where
NON-ACTIVATEDVERSIONwww.avs4you.com
118 CHAPTER 11
Example 2. Let at = t[l - (log*)~2~')(e > 0). Then fx{t) = (log*J+e and
L{fi) < oo. Since
_!_«, f = oo if 6 = 0,
we obtain
Remark 3. By Theorems 10.1, 10.2 and 10.3, we find that the statement of
Example 2 remains true replacing 77@,*) by m+(t) or M+(n). (Compare this
result with the Theorem of Hirsch of Section 5.3.)Finally we mention the following analogue of Theorem 11.9 (cf. also Theorem
11.3).
THEOREM 11.12 (Csaki - Foldes, 1986). Let at(t > 0) be a nondecreasingfunction oft satisfying conditions (i) and (ii) of Theorem 11.7. Then
liminft?tQ(*) = 1 a.s.t—»oo
where
Q{t) = inf
and
_ /log(t/q.) + loglogt'\'/'*[ )
If we also assume that (iii) of Theorem 11.7 holds then
lim &tQ{t) = 1 a.s.t»oot—»oo
Remark 4. In case at = t we obtain A1.3) as a special case of Theorem 11.12.
Remark 5. The study of the increments of r}(x,t) in x or in both variables looks
a challenging question.
11.3 Increments of
In Section 10.1 we have seen that the strong theorems valid for r}(x, t) resp. rj(t)remain valid for f(z, n) resp. f(n) due to the Invariance Principle (Theorem
NON-ACTIVATEDVERSIONwww.avs4you.com
STRONG THEOREMS OF THE LOCAL TIME 119
10.1). In Section 7.3 we have seen that the strong theorems proved for the
increments of a Wiener process remain valid for those of a random walk if an ;»
logn resp. an » (lognK depending on what kind of theorems we are talkingabout. This latter fact is due to the Invariance Principle 1 (Section 6.3) and
especially the rate O(logn) in it. Since the rate in Theorem 10.1 is much worse
(it is ©(n1/4"*"') only) we can only claim (as a consequence of the Invariance
Principle) that the results of Section 11.2 remain valid for ?(x, n) (instead of
rj(x,t)) if an > n1/2�"'. The case an < n1/2�"' requires a separate study. This
was done by Csaki and Foldes A984/C). They proved that Theorem 11.7 remain
valid for ?(x, n) if an » logn. In fact they proved the following two theorems:
THEOREM 11.13 Let 0 < an < n(n = 1,2,...) be an integer valued nonde-
creasing sequence. Assume that an/n is nonincreasing and
Then
lim -—— = oo.n—oo log n
Iimsup5n sup {?{x, k + an) — ?{x, k)) = 1 a.s.
n—oo 0<k<n-an
If we also have
log(n/an)lim -—-r1—- = oo
n—oo log log n
then
lim 6n sup {?{x,k + an) — ?(x,k)) = 1 a.s.n^°°
0<k<n-an
for any fixed x 6 Z1 where
6n = a'1'2 (log(nO + 21oglogn)�/2.THEOREM 11.14 Let c> 0. Then for any fixed x € Z1
f(z,/c + [clogn]) - ?{x,k)lim max r— : = a[c) a.s.
n—ooo<*<n-[<;logn] [C log n]
where a(c) = 1/2 if c < (Iog2)-1 and the only solution of the equation
- = A - 2a) log(l - 2a) - 2A - a) log(l - a)C
ifc>
NON-ACTIVATEDVERSIONwww.avs4you.com
120 CHAPTER 11
Remark 1. The above theorem suggests the conjecture:
=[|] a.S.
for all but finitely many n provided that an — o(logn).Since the Invariance Principle 1 of Section 6.3 is valid with the rate O(logn)
Theorem 11.8 implies
THEOREM 11.15 Theorem 11.13 remains true replacing ?{x,n) by M+(n).
The analogue of Theorem 11.9 for f(z, n) is unknown except if an > n
The analogues of Theorems 11.10 and 11.11 can be obtained by the Invariance
Principle for ?(x, n).
11.4 Strassen type theorems
Let
xt) @<x<l,t>0)
and
Un(x)=bn[t[0,k)+n(x-±)(Z@,k + l)-Z@,k))) if ?<*<^(k = 0,1,2,..., n — 1; n = 1,2,...). We intend to characterize the limit points of
the sequence Un(x) and those of ut(x). Since Un(x)@ < x < 1) for any fixed n is
a nondecreasing function, its limit points must also be nondecreasing.
Definition. Let Sm C S be the set of nondecreasing elements of S (cf. Notations
to the Strassen type theorems).Then we formulate
THEOREM 11.16 (Csaki - Revesz, 1983). The sequence {Un(x);0 < x < 1}and the net {ut(x);0 < x < 1} are relatively compact in C@,1) with probability1 and the sets of their limit points are S
Proof. This result is a trivial consequence of Theorems 8.2 and 10.2.
Define the process p(xn) @ < x < l;n = 1,2,...) by p(xn) = p* if x =
k/n (k = 0,1,2,..., n) and linear between k/n and (k + l)/n. Then taking into
account that pn is the inverse of f@, n), i.e. f@, pn) = n, we obtain the followingconsequence of Theorem 11.16:
NON-ACTIVATEDVERSIONwww.avs4you.com
STRONG THEOREMS OF THE LOCAL TIME L21
THEOREM 11.17 The set of limit points of the functions
{2rT2(log log n)p{xn); 0 < x < l} (n -> oo)
consists of those and only those functions f(x) for which f~1(x) 6 Sm .
It is also interesting to characterize the sets of limit points of the sequences
?(x,n) resp. r}(x,t) when we consider them as functions of n resp. t and we
choose a big but not too big x. In fact the Other LIL (cf. Section 5.3) tells us
that
f(zn,n)=0 resp. r](xt,t) = 0 i.o. a.s.
if
resp.\og\ogt
Hence we consider the case when x is smaller than the above limits, i.e. when
f (¦, ¦) and 77(-, ¦) are strictly positive a.s. Now we formulate
THEOREM OF DONSKER AND VARADHAN A977). In the topologyof C(—oo,+oo) the set of limit points of the functions
(/\ 1/2 \
x( ] t\ (t_*oo)\loglogty /
resp.
-—: ) ,n (n -*¦ oo)\ log log nj J J
consists of those and only those subprobability density functions f(x) for which
-r dx < 1.
Remark 1. Mueller A983) gave a common generalization of the Theorem of
Donsker - Varadhan and that of Wichura (cf. Section 8.4).
NON-ACTIVATEDVERSIONwww.avs4you.com
122 CHAPTER 11
11.5 Stability
Intuitively it is clear that f (z, n) is close to f (y, n) if x is close to y. This Section
is devoted to studying this problem.
THEOREM 11.18 (Csorgo - Revesz, 1985/A).
, N) - ?@, N)_
\Z(k,N)-Z@,N)\
, n) - ?@,n)\
= 2BA: -
where k = ±1, ±2,
THEOREM 11.19 (Csorgo - Revesz, 1985/A).
a.5.
/128N1/4= (ifJ °-
Remark 1. Since for any x € Z1,
?(x, n) = f@, n) i.o. a.s.
the study of the liminf of | ?(x,n) — f@, n) | is not interesting. The limsupproperties of f(z, n) - f@, n) follow trivially from Theorems 11.7 and 11.8.
Theorem 11.18 stated that ?(x, n) is close to f@, n) for any fixed x if n is bigenough. The next two Theorems claim that in a weaker sense ?(x,n) is nearlyequal to f@, n) in a long interval around 0.
THEOREM 11.20 (Csaki - Foldes, 1987). Put
9{t)~
Then
and
lim sup ^-1
limsup sup ^-1
= 0 a.5. if p>2
> 1 a.5. if p<\.
A1.5)
NON-ACTIVATEDVERSIONwww.avs4you.com
STRONG THEOREMS OF THE LOCAL TIME 123
THEOREM 11.21 (Csaki - Foldes, 1987). Put
M+(n)log n (log log n)"' log n(log log n)"'
Then
and
lim supfW00
= 0 a.s. t/ p > -
It
lim sup suprwoo ehi(n)<x<
where c is any positive constant.
-1 = oo a.3. = 0
Remark 2. The Theorem of Hirsch says that ^(x, n) = 0 i.o. a.s. if x >
n1/2(logn)~1. Hence it is clear that A1.5) can be true only if g(n) > n1/2(logn)~1.Theorem 11.20 tells us that g(n) must be smaller than this trivial upper estimate.
Theorem 13.18 will describe the behaviour of ?(M+(n) — j,n) when j is small.
It implies that ?(M+(n) — j,n) is much, much smaller than f@, n). Theorem
11.21 gives the longest interval, depending on Af+(n) and M~(n), where ?(x, n)is stable.
In order to prove Theorem 11.18 we present a few lemmas.
LEMMA 11.1 Let
i = ai{k) = ?{k, Pi) - ?{k, pi-d - 1 (i = 1,2,..., k = 1,2,...).
Then
and
Eax = 0, Ea2 = 4k - 2,
Jim P {n�/2(a1(A:) + at[k) + ¦¦¦ + an{k)) < x{4k - 2I/2}= B7T)-1/2 r e~u2/2du, -oo<x<oo,
J—oo
lim P In-1'2 sup(a1(A:) + a2{k) +•¦¦ + ay (A:)) < x{4k - 2I/21n^°° I J<n J
= {*)V* [*c-»V*du, x>0,7T JO
lim S&
A1.6)
A1.7)
A1.8)
(u.9,
NON-ACTIVATEDVERSIONwww.avs4you.com
124 CHAPTER 11
Proof. A1.6) is a trivial consequence of Theorem 9.7. A1.7), A1.8) and A1.9)follow from Theorems 2.9, 2.12 and the LIL of Khinchine of Section 4.4 respec-
respectively.The following two lemmas are simple consequences of A1.9).
LEMMA 11.2 Let {//„} be any sequence of positive integer valued r.v.'s with
limn_oo //„ = oo a.s. Then
«!(*)+«,(*) + .. +«,.(*)_ 1/2
rwoo (fin log log fiI'2
LEMMA 11.3 Let {vn} be a sequence of positive integer valued r.v.'s with the
following properties:
(i) lim^oo vn = oo a.s.
(ii) there exists a set flo C fl such that P(ft0) — 0 and for eac^ w ^ fl0 and
k = 1,2,... there exists an n = n(w, k) for which vn{u>,k) = k.
Then
limsup°lW t
n-oo (^
Utilizing Lemma 11.3 with un = ^@, n) and the trivial inequality ot\[k) +a.z[k) +
... + ac@,n)(A:) < ^(/c,n) - ?@, n) < a^/c) + a2[k) +... + aC(o,n)+i(A:) + 1, we
obtain Theorem 11.18.
As far as the proof of Theorem 11.19 is concerned we only present a proof of
the statement
,JV)-{(O,AT) rt28\'"
The other statements of Theorem 11.19 are proved along similar lines.
The proof of Theorem 11.19 is based on the following result of Dobrushin
A955).
THEOREM 11.22
Remark 3. One can also prove that (cf. Theorem 12.1 and A2.17))
NON-ACTIVATEDVERSIONwww.avs4you.com
STRONG THEOREMS OF THE LOCAL TIME 125
This fact together with Theorem 11.22 implies that the rate in the uniform
Invariance Principle (Theorem 10.1) cannot be true with rate n1/4.Dobrushin also notes that if N\ and N2 are independent normal @,1) r.v.'s
then the density function g of \N\\ll2Ni is
2 f°° ( v2 z4\ .
Hence Theorem 11.22 can be reformulated by saying that
(n -> oo). A1.10)
In fact this statement is not very surprising since on replacing n by f@, n)and k by 1 in A1.7), intuitively it is clear that
0~) ,. „
^^
To find an exact proof of A1.11) is not simple at all. We will study this
question in Chapter 12.
Also, by Theorem 9.12
«�/4(e(O,n)I/2 4 W1'2 (n -*. oo). A1.12)
Intuitively it is again clear (for an exact formulation see Chapter 12) that
are asymptotically independent. A1.13)
Hence A1.11), A1.12) and A1.13) together imply A1.10). The proof of Dob-
Dobrushin is not based on this idea. Following his method, however, a slightlystronger version of his Theorem 11.22 can be obtained.
THEOREM 11.23 Let {xn} be any sequence of positive numbers such that
xn = o(logn). Then
^ '
7C J—oo JO
and
P
NON-ACTIVATEDVERSIONwww.avs4you.com
126 CHAPTER 11
The following lemma describes some properties of the density function g(y). Its
proof requires only standard analytic methods, the details will be omitted.
LEMMA 11.4
(i) There exists a positive constant C such that for any y ? R1
A1.14)
(it) For any e > 0 there exists a C = C[e) > 0 such that
(iii) Let {an} be a sequence of positive numbers with an j oo. Then for any
e > 0 there exist a C\ = Ci(e) > 0 and a C2 = C2{e) > 0 such that
By Theorem 11.23 and (iii) of Lemma 11.4 we have
LEMMA 11.5 For any e > 0 there exist aCx = Ci{e) > 0 and a C2 = C2(e) >
0 such that
P |n-^(e(l,0)- ?(<),»)) > A + 2e) [^)
'
(loglognK/4| < C,(logn)-<1+'>
and
A28\ *f*
^))>(l-2e)( —) (loglogn
Now we prove
LEMMA 11.6
i) . /128N1/4
NON-ACTIVATEDVERSIONwww.avs4you.com
STRONG THEOREMS OF THE LOCAL TIME 127
Proof. Let
nk = (exp(fclogfc)],
f(n) = ?(l,n)-?@,n),, (m, n)) = ?{x, n) - ?{x, m) [m < n),
A* = U{nk)>{l-2e)dk},fa = {l-2e)dk.
By Lemma 11.5
P{At} > C{k\ogk)-^-'\ A1.15)Let y < fc and consider
oo
— A — V* V* P/ At. <-(n A — I 9 — t\
= E EPM* UK) = '.5-
00
00
< E p{?(«*) > /?* - np{f(
t | On. — X
nj) = l}
M = 1, snj =.
.P{?(n,-) = /, Sn
/ / P { f(ij) =
X
,= *}
>Pk- 21/2nJ1/4y}P{f(ni) = 21/2nJ1
/
where
A = %2-WnJ1'* = A - 2*J�/' (^) (log log
and
NON-ACTIVATEDVERSIONwww.avs4you.com
128 CHAPTER 11
(log log**K'" -
y (=*Now a simple but tedious calculation shows that for any e > 0 there exists a josuch that if j0 < j < k, then
?{AjAk} < (l + e)F{Aj}F{Ak}. A1.16)
Here we omit the details of the proof of this fact, and sketch only the main
idea behind it. Since (nj/n*I/4 < Ar1/4^" = 1,2,...,A; - l), the lower limit of
integration B(y) above is nearly equal to
(^) (loglogn*K/4 if y<k'l\ say.
Hence for latter y values the integral fj?y) g(z)dz is nearly equal to T*{Aic}. Sim-
Similarly, the integral /JJ° g{y)dy gives ~P{Aj}, and A1.16) follows, for in the case of
y > A;1/4 the value of g(y) is very small.
Now A1.15), A1.16) and the Borel - Cantelli lemma combined give Lemma
11.6.
We have also
LEMMA 11.7 Let
mk = [exp(A;/log2A;)]and
Bk = {?@,G71*, m*^)) > ak+1}where
{mk+1 - mk) (log — + 2 log log 771*+!) .
V rnk+i-
m* J\-
m*
Then of the events Bk only finitely many occur with probability 1.
Proof. This lemma is an immediate consequence of Theorem 11.13.
LEMMA 11.8 Let
M*+i = (B + eO7i*+i loglogm*+iI/2and
Dk = I sup sup |a/(l) + a/+1(l) H + a1+y(l)|{l<Mk+i-ak+1 j<ak+1
> V2 [B + e)ak+1 (log^ + \og\ogMk+1^ |Then of the events Dk only finitely many occur with probability 1.
NON-ACTIVATEDVERSIONwww.avs4you.com
STRONG THEOREMS OF THE LOCAL TIME 129
Proof. Cf. Theorem 7.13.
A simple consequence of Lemmas 11.7, 11.8 and Theorem 11.23 is
LEMMA 11.9 Let
Ek = l sup ^(m^n)! > 2B + er)ajfc+i flog—^+loglogMJfc+ij| >.
Then of the events Ek only finitely many occur with probability 1.
LEMMA 11.10
limsuP -T7771—;—hn -̂
V27/
Proof. Let
\ LI /
Then by Lemma 11.5 only finitely many of the events Fk occur with probability1. Now observing that
Mk+l M1/21 I}
log—^1+loglogMJfc+1)| = o{ck),ak+i J\
we have A1.17) by Lemma 11.9 and Lemma 11.10 is proved.Also Lemmas 11.6 and 11.10 combined give Theorem 11.19.
11.6 Favourite points
The random set Jn = {x : ?{x,n) = f(n)} will be called the set of favourite
points of the random walk {S(n)} at time n. The largest favourite points will
be denoted by fn = max{i: x G /„}.Of the properties of {/„} it is trivial that fn < u(n) with probability 1 except
for finitely many n if u(n) G UUCEn). Hence we have a trivial result sayingthat fn cannot be very large. The next theorem claims that /„ occasionally will
be large.
THEOREM 11.24 (Erdos - Revesz, 1984). For any e > 0
with probability 1 infinitely often.
NON-ACTIVATEDVERSIONwww.avs4you.com
130 CHAPTER 11
Having this result, one can conjecture that /„ will be larger than any function
l[n) i.o. with probability 1 if l(n) € ULCEn).However, it is not the case. Conversely, we have
THEOREM 11.25 (Erdos - Revesz, 1984).
fn < (nB log2 n + 3 log3 n + 2 log4 n + 2 log5 n + 2 log6 n
with probability 1 except for finitely many n.
It looks also interesting to investigate the small favourite points. Let gn =
min{|x| : x 6 /„}. Bass and Griffin A985) proved that gn cannot be very
small. In fact
THEOREM 11.26
joo if 1>U,
)--* \0 if -y<2.
Here we present a few unsolved problems (Erdos - Revesz, 1984 and 1987).
1. Theorem 11.24 stated that /„ > (I — e)b~1 infinitely often with probability1. Its proof shows that when /„ > (l — e)b~1, then f (/„, n) = f (n) will be
larger than Db~l (where D is a small enough positive constant) infinitelyoften with probability 1. It is not clear how big f(n) can be when /„ >
A — e)b~1 or how big /„ can be when f (n) > A — ^b'1.
2. Everyone can see immediately that \7n\ = 1 and \7n\ = 2 i.o. with proba-probability 1. Can we say that |/n| > 3 infinitely often with probability 1?
3. Consider the random sequence {un} for which |/i,n| > 2. What can we say
about the sequence {^n}? Can we say, for example, that limn—oo vn/n = oo
with probability 1?
4. How can the properties of the sequence |/n+i — fn\ be characterized? Is it
true that limsup,^^ |/n+i — fn\ = oo? If yes, what is the rate of conver-
convergence?
5. Does the sequence /n/\/n have a limit distribution? If yes, what is it?
6. Let ot(n) be the number of different favourite values up to n, i.e. a(n) =
| Z))b=i ?k\- We guess that ct(n) is very small, i.e. a(n) < (logn)c for some
c > 0, but we cannot prove it. Hence we ask: how can one describe the
limit behaviour of a(n)?
NON-ACTIVATEDVERSIONwww.avs4you.com
STRONG THEOREMS OF THE LOCAL TIME 131
7. We also ask how long a point can stay as a favourite value, i.e. let 1 < i =
*(n) < j — ]{n) < n be two integers for which
and j — i = /?(n) is as big as possible. The question is to describe the limit
behaviour of /?(n).
8. Further if x was a favourite value once, can it happen that the favourite
value moves away from x but later returns to x again, i.e. do sequences
an < bn < cn of positive random integers exist such that
?„/>„= 0 and /an/Cn^0 (n = l,2,...)?
9. To investigate the jumps of the favourite values looks also interesting. Let
n = n(w) be a positive integer for which 7n?n+i — 0- Then the jump-sizejn is defined as
jn = p(?., ?.+i) = min{|x-y|; x € /n,y € /n+i}.
The theorem of Bass and Griffin (Theorem 11.26) implies that jn >
n1/2(logn)~11 i.o. a.s. It looks very likely that limn—oojn = oo a.s. We do
not see how one can describe the limit behaviour of jn.
10. By the arcsine law we learned that the particle spends a long time on one
half of the line and only a short time on the other half with a big probability.We ask whether the favourite value is located on the same side where the
particle has spent the long time. For example let 0 < ti\ < ni(w) < n-i...
be a random sequence of integers for which
where
J 1 if S' > 0
Then we conjecture that /nt —*• oo as A; —*• oo a.s.
NON-ACTIVATEDVERSIONwww.avs4you.com
132 CHAPTER 11
11.7 Rarely visited points
It is easy to see that for infinitely many n almost all paths assume every value
at least twice which they assume at all, i.e. let 6^ = 0 if f@, n) ^ r - 1 and
<$ir) = 1 if f@,n) = r - 1 and let
be the number of points visited exactly r-times up to n. Then
P{/1(n)=0i.o.} = l.
We do not know if for infinitely many n almost all paths assume every value at
least r-times (r = 2,3,...) which they assume at all, i.e. let
and we ask
P{(/r(n)=0i.o.} = ?
We would guess that this probability is 0 if r > 2 but perhaps it is 1 if r = 2.
A study of
liminf/r(n) and limsup/r(n) (r = l,2,...)n-K» n-K»
looks also interesting.As already stated liminfn—oo /i(n) = 0. Major A988) proved
THEOREM 11.27
limsupl
,= C a.s.
n—oo log n
where 0 < C < oo but its exact value is unknown.
Another interesting result on /i(n) is
THEOREM 11.28 (Newman, 1984).
E/!(n)=2 (n = l,2,...).Proof. Since /x(l) = 2 we only prove that E/^n + l) = E/Jn) for n > 0.
Consider the walk S? = Sk+i — Xi (k = 0,1,2,...) and let '/i (n) ^e t^ie number
of points visited exactly once by S? up to n. Then
/*(n) + l if ?@,n+l)=0,/i(n)-l if ?@,n+l) = l,
K{n) if e@,n+l)>l.Theorem 9.3 implies that P{^@,n + l) = 0} = P{?@,n + 1) = 1}. Hence we
have the Theorem.
NON-ACTIVATEDVERSIONwww.avs4you.com
Chapter 12
An embedding theorem
12.1 On the Wiener sheet
Let {Xij, i = 1,2,...,y = 1,2,...} be a double array of i.i.d.r.v.'s with
and define S0>n = Sm<0 = 0 (n = 0,1,2,...; m = 0,1,2,...},
7=1i=l
The arrays {S^m} and {Xij} are called random fields. Some properties of {S^m}can be obtained as simple consequences of the corresponding properties of the
random walk, some properties of {5>,,m} are essentially different. Here we men-
mention one example of both types. Just like in the one-dimensional case we have
lim p(^^ <*} = $(*).{/ Jn-~ {y/nrn
However,
Iimsup6(nmMnm = 21/2 a.s.
m—t-oo
This latter result is due to Zimmermann A972) (see also Csorgo - Revesz, 1981).On the same way as the Wiener process was defined (Section 6.2) a continuous
analogue of {Snm, n = 0,1,2,...; m = 0,1,2,...} can be defined. This contin-
continuous random field will be called Wiener sheet (two-parameter Wiener process)and will be denoted by
{W{x,y), x>0,y>0}.
Among the properties of the Wiener sheet we mention
133
NON-ACTIVATEDVERSIONwww.avs4you.com
134 CHAPTER 12
(i) W(-, •) is a Gaussian process,
(ii) W{0,y)=W{x,0)=0,
(Hi) EW(xuy1)W(x2, y2) = min(xi,x2) min(yi,y2),
(iv) VT(x,y) is continuous a.s.,
— 1/2
(v) for any x0 > 0, the one-dimensional process {xQ W{x0, y), y > 0} is a
Wiener process,
(vi) for any y0 > 0, the one-dimensional process {y0 W(x,y0), x > 0} is a
Wiener process.
For some further study and a detailed definition of the Wiener sheet we refer
to Csorgo - Revesz A981).
12.2 The theorem
We have already seen that the study of the processes ?(x,n) resp. r}(x,t) is
relatively easy when x is fixed and we let only n resp. t vary. The main reason
of this fact is the following trivial:
LEMMA 12.1 For any integer x
Z{x,Pi) - ?{x,p0) = Z[x,pi),Z{x,p2) - ?{x,pi),Z{x,p3) - ?(x,p2),...
are i.i.d.r.v.'s with
E(e(*,P*) - e(*,p*-i)) = l,E(?(z,pfc) - ?(*,p*-i) - IJ = 4x - 2
(cf. Theorem 9.7).
In order to formulate the analogue of Lemma 12.1 for rj(-, ¦) let
Po=O, p: = mf{t;t>0,V{0,t)>u} (u > 0). A2.1)
Then we have
LEMMA 12.2 For any x ? R1, r)(x,p*u) is a process of independent increments
in u(u > 0), i.e. for any 0 < ui < u2 < ... < ujt(A; = 1,2,...), the r.v. 's
NON-ACTIVATEDVERSIONwww.avs4you.com
AN EMBEDDING THEOREM 135
are independent with
where j = 1,2,..., A;.
Consider the process
?(x,u) = ^(x.pi) - 7/@,P;) = ^(x,P;) - u. A2.2)
Then we have
(i)E?(x,u)=0, E?2[x,u) =4xu,
(ii) {?(x, u);u > 0} is a strictly stationary process of independent increments
in u for any x 6 J?1.
One can also prove that
(iii) ?(x, u) has a finite moment generating function in a neighbourhood of the
origin.
By the Invariance Principle 2 (cf. Section 6.3) this fact easily implies that for
any iGi?1 the process ?(x,u) can be approximated by a Wiener process W*(-)with rate O(logu), i.e.
Having a fixed x this result gives an important tool to describe the properties of
C{x,u).What can we say about ?(x,u) when u is fixed and x is varying? It is easy
to prove that for any fixed u {?(x,u),x > 0} has orthogonal increments and it
is a martingale in x. This observation suggests the question:Can the process ?(x, u) be approximated by a two-parameter Wiener process?Since by the LIL r}(x, u) = 0 a.s. if x > (B + ejulogloguI/2 and u is big
enough, we have ?(x, u) = —u for any x big enough. This clearly shows that
the structure of ?(x, u) is quite different from that of W(x, u) whenever x is big.Hence we modify the above question as follows:
Can the process ?(x, u) be approximated by a Wiener sheet provided that u
is big but x is not very big?The answer to this question is positive. In fact we have
NON-ACTIVATEDVERSIONwww.avs4you.com
136 CHAPTER 12
THEOREM 12.1 (Csaki - Csorgo - Foldes - Revesz, 1989). There exists a
probability space with
(i) a standard Wiener process {W(t),t > 0}, its two-parameter local time pro-
process {rj(x, t),x 6 Rl,t > 0} and the inverse process p*u of 77@,*) defined by
A2.1),
(ii) a two-time parameter Wiener process {W(x,u);x > 0, u > 0} such that
sup \?{x,u) -2W[x,u)\ =o(ui?-e) a.s. [u -* 00) A2.3)0<x<Aus
where ?(x, u) is defined by A2.2), A is an arbitrary positive constant and
0 < 6 < 7/100,0 < e < 1/72 - 6/7.
This theorem is certainly a useful tool for studying the properties of ?(x, u) or
r}(x,p*u). Unfortunately it does not say too much about rj(x,t). However, we can
continue Theorem 12.1 as follows:
THEOREM 12.2 On the probability space of Theorem 12.1 we can also definea process p*u such that
{p>>0}?{p;,u>0}, A2.4)
\PI - P*u\ = O (u15/8) a.s. (u-00), A2.5)
{Pu>« > 0} and {W(x,u);x>0,u> 0} are independent. A2.6)
Having the process {p*, u > 0} we can proceed as follows:
Define the local time process rj{O,t) by
By the continuity properties of 77@,*) (cf. Theorem 11.7) we have
= 0 a.s.
Thus by Theorem 12.2 we conclude that the local time process {77@,*);* > 0}has the following properties:
{rj{O,t);t > 0}Mv{0,t);t > 0}, A2.7)
1*7@,*) - v[O,t)\ is small a.s. (t -* bo), A2.8)
{17@,t);t >0} and {W(x,u);x >0,u > 0} are independent. A2.9)
NON-ACTIVATEDVERSIONwww.avs4you.com
AN EMBEDDING THEOREM 137
A2.7) resp. A2.9) follows immediately from A2.4) resp. A2.6). In order to see
A2.8) it is enough to show that
is small, which in turn follows from the fact that
is small. Now A2.3), A2.8), and the continuity of W(-, •) imply
\rj{x,t) -77@,*) -2W(x,f)@,t))\ is small a.s.
where 77@,*) satisfies A2.8) and A2.9).A precise version of the above sketched idea implies
THEOREM 12.3 (Csaki - Csorgo - Foldes - Revesz, 1989). There exists a
probability space with
(i) a standard Wiener process {W(t);t > 0} and its two-parameter local time
process {r)(x,t);x E R},t > 0},
(ii) a two-time parameter Wiener process {W(x,u);x > 0,u > 0},
(iii) a process {f){O,t);t >0}={r}{0,t);t > 0}
such that
sup \n{x,t)-r,{0,t)-2W{x,f,{0,t))\=o(ti?->) a.s. (t -» 00),<AT*/2
a.s. (t -> 00),
{17@, t); t > 0} and {W(x, u); x > 0, u > 0} are independent
where
A > 0,0 < 6 < 7/100, 0 < e < 1/72 - 6/7.
12.3 ApplicationsIn order to show how the above theorem can be used in the study of the propertiesof 77(-, •), first we list a few simple properties of the vector valued process
NON-ACTIVATEDVERSIONwww.avs4you.com
138 CHAPTER 12
which can be obtained by standard methods of proof.Namely for any x > 0 and t > 0 we have
where Ni,N2 are independent normal @,1) r.v.'s.
Also, for any x > 0, the set of limit points of
Ut=
is the interval [—1,1] a.s. The set of limit points of
y
t-l'*f)(O,t) I \N2\, A2.11)
llNl\N^ A2.12)
y/2t log log t
is the interval [0,1] a.s. The set of limit points of
(Ut,Vt)
is the semidisc {{u,v) : v > 0,u2 + v2 < 1}. The set of limit points of
rri/i/2 w(M(o,0)
is the interval [0,21/23�/4] a.s. for any x > 0, that is,
The usual LIL implies
i- W(x,fj@,t))hmsup sup . ===== = 1 a.s.
«-»«, o<z<Kt* yj2Kri @, t)ts log log t
Applying again the independence of Ut and Vt we obtain
NON-ACTIVATEDVERSIONwww.avs4you.com
AN EMBEDDING THEOREM 139
for any K > 0 and 6 > 0.
Consequently, by Theorem 12.3 and by A2.10), A2.11), A2.12), A2.13),A2.14) respectively we obtain
2yJxr,{0,t)0
limsup i/iw!n?@>/?3/4 = ^6l/4 a-s-
x1/2i1/4(lltK/4 3
i- r,{x,t) - r,{O,t)limsup sup
*(x,t) - r,(O,t) ± ^ forany x>Q ^^
a| (t >0), A2.16)
^\1'2 *->o0' forany
limsup—yifL=L===2LjJ== = 1 a.s. forany x > 0, A2.18)*-«» 2yJ2xr)@, t) log log t
sup ,=
0<x<Kt* 2^/2Ktsri@, t) log log t
i-3 1/4 n(x,t) - r}(O,t)
limsup sup -6-1'* . ,' /.n> j
/4=1 a.s. 12.20
for any /f > 0 and 0 < 6 < 7/200.For the direct proofs of A2.18) and A2.19) see Csaki - Foldes A988).
NON-ACTIVATEDVERSIONwww.avs4you.com
Chapter 13
Excursions
13.1 On the distribution of the zeros of a
random walk
(9.11) and Theorem 11.1 are telling us in different forms that f@, n) convergesto oo like n1/2, i.e. the particle during its first n steps visits the origin practicallyn1/2 times. Clearly these n1/2 visits are distributed in [0,n] in a very nonuniform
way. We have already met the Chung - Erdos theorem (Theorem 11.10) and
the arcsine law (9.12) claiming that the zeros of {Sk} are very nonuniformlydistributed at least for some n. Now we give a few reformulations of the Chung- Erdos theorem in order to see how it describes the nonuniformness of the
distribution of the zeros of {Sk}. First a few notations:
(i) let
R{n) = max{A; : k > 1 for which there exists a 0 < j < n - k
such that f@,y + A;) - f@, j) = 0}
be the length of the longest zero-free interval,
(ii) let
R{n) = max{A;: k > 1 for which there exists a 0 < j < n - A;
such that M+{j + k)= M+{j)}
be the length of the longest flat interval of M? up to n,
(iii) let
#(n) = max{Jt : 1 < k < n, Sk = 0}be the location of the last zero up to n,
141
NON-ACTIVATEDVERSIONwww.avs4you.com
142 CHAPTER 13
(iv) let <;n be the number of those terms of 5j, 52,..., Sn which are positive or
which are equal to 0 but the preceding term of which is positive,
(v) let
fi+(n) = inf{Jt : 0 < k < n for which Sk = M+}.
Now we can reformulate the Chung - Erdos theorem (Theorem 11.10) as
follows:
THEOREM 13.1 Let f(x) be a nondecreasing function for which lim^oo f(x)= oo, x/f(x) is nondecreasing and lim^oo x/f(x) = oo. Then
if and only ifr°° dx
h
where Y(n) is any of the processes R(n),R(n),n — \&(n),?n,n — fi+(n).
Proof. It is immediately clear that
UUC(i?(n)) = UUC(n - *(n))
and
UUC(JR(n)) = UUC(n - fi+{n)).
By Theorem 10.3 it is also clear that
UUC(J2(n)) = UUC(^(n)).
As far as the process $n is concerned we clearly have
Cn) C UUC(n - V{n)).
The equality in the last relationship is not quite clear but following the originalproof of Theorem 11.10 given by Chung and Erdos A952) we get the requiredequality.
The characterization of the lower classes of n — \&(n) is trivial since we have
= n i.o. a.s.
The characterization of the lower classes of $n is also trivial. In fact as a simpleconsequence of Theorem 13.1 we obtain
NON-ACTIVATEDVERSIONwww.avs4you.com
EXCURSIONS 143
THEOREM 13.2 Assume that f(x) satisfies the conditions of Theorem 13.1.
Then
7^7 € LLC(fn)
i/ and on/y i/
z(/(z))i/'<0°-
The characterization of the lower classes of i2(n) and R(n) is much harder. We
have
THEOREM 13.3 (Csaki - Erdos - Revesz, 1985). Let f(x) be a nondecreasingfunction for which
/(x)/oo,—/oo (z->oo).
Then
if and only if
n=l
where Y* is any of the processes R(n) and R(n) and E = 0,85403... is the root
of the equationoo
yk=l
Consequence 1.
. log log n. log log nblim inf R(n) = lim inf R(n) = 3 a.s.()
Besides studying the length of longest excursion R(n), it looks interesting to say
something about the second, third,... etc. longest excursions. Consider the sam-
sample Pi,p2-
Pi, ••• iP€(o,n) -Pc(o1n)-n»»-Pc(o1n) (the lengths of the excursions) and
the corresponding ordered sample Ri{n) = R(n) > R2{n) > ... > i?f@,n)+i(n).Now we present
THEOREM 13.4 For any fixed k = 1,2,... we have
NON-ACTIVATEDVERSIONwww.avs4you.com
144 CHAPTER 13
This theorem in some sense answers the question: How small can R2(n),R3(n),...be? In order to obtain a more complete description of these r.v.'s we present the
following:
Problem 1. Characterize the set of those nondecreasing functions /(n) (n =
1,2,...) for which
Theorem 13.1 tells us that for some n nearly the whole random walk {S(k)}%=0is one excursion. Theorem 13.3 tells us that for some n the random walk consists
of at least 0~l log log n excursions. These results suggest the question: For what
values of k = k(n) will the sum Z)*=1-R;-(n) be nearly equal to n? In fact we
formulate two questions:
Question 1. For any 0 < e < 1 let T(e) be the set of those functions f(n) (n =
1,2,...) for which()
with probability 1 except finitely many n. How can we characterize /(e)?
Question 2. Let J{o) be the set of those functions f{n)(n = 1,2,...) for which
lim n1 y -R.-(n) = 1 a.s.
n—»oo*—' ' v '
How can we characterize /(o)?Studying the first question we have
THEOREM 13.5 (Csaki - Erdos - Revesz, 1985). For any 0 < e < 1 there
exists a C = C(e) > 0 such that
Clog logn€ J{e).
Concerning Question 2, we have the following result:
THEOREM 13.6 For any C> 0
/(n) = C log log n$f{o)
and for any h(n) /" oo (n —> oo)
/i(n) loglogn € J(o).
NON-ACTIVATEDVERSIONwww.avs4you.com
EXCURSIONS 145
Knight A986) was interested in the distribution of the duration of the longestexcursion of a Wiener process. In order to formulate his results introduce the
following notations: for arbitrary i>Owe set
to(t) = sup{s : s < t,W(s) = 0},
ti(t) = inf{s : s > t,W(s) = 0},
d{t) =*!(*) -to{t),D{t) = sxxp{d{s) : to(s) < t},
E(t) = snp{d(s) : s < t,ti{s) < t}.
Then we call d(t) the duration of the excursion containing t. D(t) resp. E(t) is
the maximal duration of excursions starting by t resp. ending by t.
Knight evaluated completely the Laplace transforms of the distributions of
D(t) and E(t) and the distributions themselves over a finite interval. His results
run as follows:
THEOREM 13.7 (Knight, 1986).
where
Ft ^_/27r"^1/2 «/ y^1'
*W-\x-l(Z-y+j.\Ogy) if l<y<2
and
where G{1) = 0,
if i<y<2,if 2<y<3,
and
GB) = I - i.^7T 2
The multiple Laplace transform of D(t) and some other characteristics of a
Wiener process were investigated by Csaki - Foldes A988/A). A very different
characterization of the distribution of the zeros of {Sn} is due to Erdos and
Taylor A960/A), who proved
NON-ACTIVATEDVERSIONwww.avs4you.com
146 CHAPTER 13
THEOREM 13.8
1n
lim y^Pt = T~ as.»-"» log n
~
Remark 1. (9.8) and Theorem 11.6 claim that pk converges to infinity like k2.
However, these two results are also claiming that the fluctuation of k~2pk can be
and will be very big. Theorem 13.8, via investigating the logarithmic density of
pj[ , also tells us that pk behaves like k2.
Let us mention a result of Levy A948) that is very similar to the above
theorem.
THEOREM 13.9
lim : >,
= - a.s.«-»«» log n ^ k 2
wherefl •/ Sk >0,
-{o ,/ sk<o.
Remark 2. Theorems 13.1 and 13.2 imply that
1n
liminf-V I(Sk) = 0 a.s.
and1
n
lim sup— ^2 I{Sic) = 1 a.s.
noo ftftfc=1
Hence the sequence /(S*) does not have a density in the ordinary sense but byTheorem 13.9 its logarithmic density is 1/2.
It is natural to ask what happens if in Theorem 13.9 the indicator function
/(•) of (—oo,0) is replaced by the indicator function of an arbitrary Borel-set of
R1. We obtain
THEOREM 13.10 (Brosamler, 1988; Fisher, 1987 and Schatte, 1989). There
is a P-null set N C fl such that for allu <? N and for all Borel-sets A C R1 with
X(dA) = 0 we have
lim
NON-ACTIVATEDVERSIONwww.avs4you.com
EXCURSIONS 147
where dA is the boundary of A and
For a Strassen type generalization of Theorem 13.10, cf. Brosamler A988) and
Lacey - Philipp A989).For the sake of completeness we also mention
THEOREM 13.11 (Weigl, 1989).
J \ (?,k->I(Sk) - ilogn) < xl = *(x)
where
/I fo
/(•) is defined in Theorem 13.9.
13.2 Local time and the number of longexcursions (Mesure du voisinage)
The definition of the local time of a Wiener process (cf. Section 9.3) is extrinsic
in the sense that given the random set At = {t : 0 < t < T,W(t) =0} one
cannot recover the local time rj(O,T). Levy called attention to the necessity of
an intrinsic definition.
He proposed the following: Let N(h,x,t) be the number of excursions of
W(-) away from x that are greater than h in length and are completed by time
t. Then the "mesure du voisinage" of W at time t is \imhs^0 h1?2N(h, x,t), and
the connection between rj and N is given by the following result of P. Levy (cf.Ito and McKean 1965, p. 43).
THEOREM 13.12 For all real x and for all positive t we have
lim h^2N{h, x, t) = \\-ri{x, t) a.s.
h,"\0 y 7T
Perkins A981) proved that Theorem 13.12 holds uniformly in x and t. Cso^goand Revesz A986) proved a stronger version of Perkins' result. Their results can
be summarized in the following four theorems.
NON-ACTIVATEDVERSIONwww.avs4you.com
148 CHAPTER 13
THEOREM 13.13 For any fixed t' > 0 we have
hl'2N(h,x,t)-J-r,(x,t)V 7T
sup(x,t)eRlx[O,t'}
= 0 a.s.
The connection between N and 77 is also investigated in the case when a Wiener
process through a long time t is observed and the number of long (but much
shorter than t) excursions is considered. We have
THEOREM 13.14 For some 0 < a < 1 let 0 < at < ta(t > 0) be a nonde-
creasing function oft so that at/t is nonincreasing. Then
Tlim — log —
sup<-°° \t J \ aj XfzRi
N{at,x,t)-J r,{x,t) = 0 a.s.
The proofs of Theorems 13.13 and 13.14 are based on two large deviation type
inequalities which are of interest on their own.
THEOREM 13.15 For any K > 0 and t' > 0 there exist a C = C{K,t') > 0
and a D = D{K,t') > 0 such that
sup >C\ < DhK,
where h < t'.
THEOREM 13.16 For any K > 0 there exist aC = C(K) > 0 and a
D = D{K) > 0 such that
log -at
sup N{at,x,t)-J r,{x,t) >C
where 0 < at < t.
It is natural to ask about the analogues of the above theorems for random walk.
Clearly for any x = 0,±1,±2,... the number of excursions away from x
completed by n is equal to the local time f(x, n), i.e.
M(x, n) = {the number of excursions away from x completed by n} =
max{t : pi{x) < n} = f(x, n).Hence we consider the following problem: knowing the number of long excur-
excursions (longer than a = an) away from x completed by n, what can be said
about f(x,n)? Let M(a,x,n) be the number of excursions away from x longerthan a and completed by n. Our main result says that observing the sequence
n,x,n)}^L1 with some an = [na]@ < a < 1/3) the local time sequence
n)}^=l can be relatively well estimated. In fact we have
NON-ACTIVATEDVERSIONwww.avs4you.com
EXCURSIONS 149
THEOREM 13.17 Let an = \na\ with 0 < a < 1/3. Then
/a\ 1/4 / n\-llim — 1 (log—I sup \M(an,x,n) - ?(x,n)P(an)\ = 0 a.s.n->°° V n / V an) x€Zi
where
P[a) = P{pj > a}.
The proof of this theorem is based on
THEOREM 13.18 For any K > 0 there exist a C = C(K) > 0 and a D =
D(K) > 0 such that
Kn\ 1/4 / _ \ -3/4 1
-^ log— sup \M{an,x,n) - Z{x,n)P{an)\ >C\< Dn~K,n / \ an/ X?zl )
where an = [na] @ < a < 1/3).
Remark 1. Very likely Theorems 13.17 and 13.18 remain true assuming onlythat 0 < a < 1.
In order to prove Theorem 13.18, first we prove the following simple
LEMMA 13.1 Let n and a be positive integers and C > 0. Then
M{a,x,pn{x)) -nP(a)(nP(a)(l-P(a))lognP(a)I/2 >C1/2|<2(P(a)n)-2C/9,
provided that
Clog(nP(a)) A3.1)
Proof. Clearly M(a, x,pn(x)) is binomially distributed with parameters n and
P(a). Hence the Bernstein inequality (Theorem 2.3 ) easily implies Lemma 13.1.
Proof of Theorem 13.18. Since by (9.10)
.-i'
condition A3.1) holds true if a < np(p < 2) and n is big enough, Lemma 13.1
can be reformulated as follows: for any K > 0 and 0 < if) < p < 2 there exist a
C = C{tp,p,K) > 0 and D = D{^,p,K) > 0 such that
M{a,x,pn{x)) - Z{x,pn{x))P{a)2 \
—)7ra/
>C Dn~KA3.2)
NON-ACTIVATEDVERSIONwww.avs4you.com
150 CHAPTER 13
provided that n* < a < np.
A3.2) in turn implies
supM{a,x,pn{x)) -
>C \ < Dn~K, A3.3)
and for any K > 0, 0 < V < P < 2 and 0 < 7 < <5 < 00 there exist a
C = C{i,6,tp,p,K) and a ?> = D(t,6,ip,p,K) such that
sup supM{a,x,pn{x)) - Z{x,pn{x))P{a)
„!/*(-?-)'
(logna
y>C -K. A3.4)
Then by a slight generalization of (9.11) (or applying the exact distribution of
f@,n), cf. Theorem 9.3) for any K > 0 there exist a C = C(K) > 0 and a
D = D{K) > 0 such that
Clogn<Dn~K A3.5)
for any x € Zl or equivalently
P{f(z,n) >C(nlognI/2} < fln"*. A3.6)
Let m be a fixed positive integer and assume that the event
Am = [m? < ?{x,m) < C(mlogmI/2} @ < /? < 1/2)
holds true. Then replacing m by pn(x) (more exactly assuming that f (x,m) = n,
i.e. pn{x) < m < pn+i(x)) we obtain
J = J(m,x) =M(am,x,m) - ?{x,m)P{am) M{am,x,pn{xj) - nP(am)
where by the assumption that Am holds true we have
m? < ^(x,m) = n < C(m log mI/2,
i.e.
2C2 log n
m
NON-ACTIVATEDVERSIONwww.avs4you.com
EXCURSIONS
Hence
151
J < supmP<n<C(m log mI/2
M(am,x,pn(x)) -nP(am)1/4
2C2 log n
-1/4log
n3/4
am2C2logn
< 4 sup supM(a,x,pn(x)) -nP(a)
Observe that if f (x, m) < m^ then
J <mh
Consequently
0 if 0<
=0
aJ
1-a
A3.7)
A3.8)if m is big enough and /? < A - a)/4. Hence by A3.8), A3.7) and A3.4) we
obtain
P{J >C} = P{J > C, Am} + P{J > C, f (x, m) < mp)+ P{J > C,^(x,m) > C(mlogmI/2} < P{J > C, Am}+ P{^(x,m) >C(mlogmI/2}
< P{J > C, Am} + Dm~K < 2Dm~K
if m is big enough, /? < ^-^ and | < 2. /? can be chosen in such a way if
0 < a < 1/3. Consequently we have also that
-KP{sup J(m,x) > C} < Dm
for any K > 0 if C, D are big enough and 0 < a < 1/3. Hence the proof of
Theorem 13.18 is complete.Theorem 13.17 is a trivial consequence of Theorem 13.18.
Note that if a > 1/5 then P(a) can be replaced by B/7raI/2. Hence we also
obtain
THEOREM 13.19 Let an = [na] with 1/5 < a < 1/3. Then
nsupz€Zl
M(onii,n)-((i,n) )\nan/
J/2= 0 a.s.
NON-ACTIVATEDVERSIONwww.avs4you.com
152 CHAPTER 13
THEOREM 13.20 For any K > 0 there exist a C = C{K) > 0 and a D =
D(K) > 0 such that
( /a \ !/4 / n \ ~3/4 / 2 \P (-) (log-) sup M(an,s,n)-f(s,n)( )
w/icrc an = \na\ A/5 < a < 1/3).
> C < Dn~KJ
13.3 The local time of high excursions
Theorem 9.7 described the distribution of the local time f (A:,pi) of the excursion
{S0,Si,... ,SPl}. Now we are interested in the properties of f(A:,pi) when k is
big, i.e. when k is close to M+(pi), the height of the excursion {So, Si,..., SPl}.We are especially interested in the limit distribution of f(A:,pi) when k is close
to M+(pi) = n and n —> oo. First we present the simple
THEOREM 13.21 For any n = 1,2,... and / = 1,2,... we have
P{M+(Pl) = n, f(n,Pl) = /} = n-22-'-
if n ^ oo then
= n} = 5±i (^i) - 2"',
Proof. By Lemma 3.1 the probability that the excursion {So, Sx,..., SPl} hits n
is Bft)-1, i.e. P{M+(pj) > re} = Bft)-1. The probability that after the arrival
time Pi(n) the particle turns back but hits n once more before arriving at 0 is
1/2A — 1/n). Hence the probability of / — 1 negative excursions away from n
before px is A/2A - 1/n)I�. Finally Bft)-1 is the probability that after / - 1
excursions the particle returns to 0.
In order to study the properties of ?(M+(p\) — j,pi), first we investigate the
distribution of f(M+(pj) - j,Pi{M+(pi))). (Note that p1(M+(p1)) is the first
hitting of the level M+(p\).)
NON-ACTIVATEDVERSIONwww.avs4you.com
EXCURSIONS 153
LEMMA 13.2 For any I = 1,2,..., n = 1,2,..., j = 1,2,... ,n- 1
(Ww h
-i) V 2j{n-j)
and if n —> oo t/icn
n ( n
1
2j(n-j)\ 2j{n-j)J 2j \ 2jJ
> n) = > 2j,n
— \M+{px)>n
n n
Proof.
z n —
Further,
is the probability that after p\{n — j) the particle makes u negative excursions
away from n—j (none of them reaches 0) and / — 1 — u positive excursions away
from n—j (none of them reaches n) in a given order. Finally By)-1 is the
probability that after the / — 1 excursions the particle goes to n.
LEMMA 13.3 For any n = 2,3,..., / = 2,3,..., j = 1,2,..., n - 1
(« -i.Pi) = I I A/+(p,) = n,e(n,pi) = 1} = P{?/, + U2 = 1}
4?
NON-ACTIVATEDVERSIONwww.avs4you.com
154 CHAPTER 13
where U\ and Ui are i.i.d.r.v.'s with
m-1
Further,
and
n
,...). A3.9)
(n^oo)
•n\ 2
n
n n
Proof. Since
and by Lemma 13.2 the conditional distribution of f(n — j,p\(n)) and that of
^(« - J,Pi) ~ Z{n-J,Pi{n)) (given (M+(pi) = n,^(n,p!) = 1}) are equal to the
distribution of U\ we obtain Lemma 13.3 realizing that the two terms of righthand side of A3.10) are conditionally independent.
LEMMA 13.4 For any n = 2,3,... ,j = 1,2,... ,n- 1 and I = 0,1,2,...
l-y-1
i1-
if 1 = 0,i-i
n+1 (
n
if n —^ oo then
t/ /=0,
n+1
n(n.(n + l-jJ^2,
NON-ACTIVATEDVERSIONwww.avs4you.com
EXCURSIONS 155
E
n + 1 - j
n(n 2(n + 1 - j) n8j - 6.
Proof is essentially the same as that of Lemma 13.2.
LEMMA 13.5 For any n = 2,3,..., j = 1,2,..., n - 1, k = 1,2,... and
1 = 2,3,...
-.7,Pi) = ' I M+{Pl) = n.tfn.pi) = A:}
+ vl + v2 + ..- + vk_l + u2 =
where U\, Vy, V2,..., Vk-\,U2 are independent r.v.'s with
/ \ rn-\
i = m} = ^727^rT f 1 7r2.7A1-j) V 2j(n-j)
F{V{ =m} =
j) n-1
t = l,2;m = l,2,...)
if m = 0,
n
1- rr
nm-l
if m > 1,
n-1 n(n-l)
- 1 -j) - n
n-1 n-1
- Aj - 6) = 8j2 + {2k - 3Lj - 6(ik - 1).
NON-ACTIVATEDVERSIONwww.avs4you.com
156 CHAPTER 13
Proof. Clearly
fc-i
t{n-J,Pi) = t{n-J,pM) + E(f(n -i.P<+i(n)) - t{n-J,Piin))t=i
+ {t{n-j,Pi) ~ Z{n-j,pk{n))
where
?(« " i.PiH), e(« " i,P.-+i(»)) " ?(» " i,P.'(»)) (i = 1,2, ...,*- 1)
and
^(n-JiPi) -^(n-JiPfcH)
are independent. Lemma 13.3 tells us that the conditional distributions of f (n —
j,p\(n)) and f (n — j,pi) — ?(n —j,pk{n)) are equal to the distribution of U\ and
U2. Lemma 13.4 claims that the conditional distributions of f(n - j,pt+1(n)) -
?{n—j\pi(n)) (i = 1,2,..., k — 1) are equal to the distribution of Vj, V2,..., V^j.Hence we have Lemma 13.5.
Theorem 13.21 and Lemmas 13.3, 13.4 and 13.5 combined imply
THEOREM 13.22 For any j = 1,2,..., n - 1; n = 2,3,...
(„ + 1 - j)^ 4, + 2,LZ& + (^ l)n Vn + 1 / n(n + 1)
n - j,Pl) - Ef(n - y,Pl)J | M+(Pl) = n) - 8j2 + 4j - 6.
Further, for any j = 0,1,2,... and K > 0 there exist a C\ = C\(K,j) > 0 and a
C2 = C2{K,j) > 0 such that
P {?(" -J,Pi) > Ci log" | M+(Pl) = n} < Cjn"*
/or any a > 0 and if > 0 there exist a C\ = Ci(a,K) > 0 and a C2 —
C2{a,K) >0 such that
P{^(n-alogn,p!) >C1log2n|M+(p1) = n} < Czn'^.
We also obtain the following:
Consequence 1. For any j = 0,1,2,..., n and n big enough we have
P {f (n " i.Pi) > 6J2 + 4j + 2 | M+(Pl) =n}<±.
NON-ACTIVATEDVERSIONwww.avs4you.com
EXCURSIONS 157
Proof. By Chebyshev inequality and Theorem 13.22 we have
P {tfn -i.Pi) > A(8j2 + Aj - 6I/2 + Aj + 2\ M+(Pl) = n} < 1.
Taking A = 2j and observing that
2j (8j2 + Aj - 6I/2 + Aj + 2 < 6j2 + Aj + 2
we obtain the above inequality.
13.4 How many times can a random walk
reach its maximum?
Let x{n) be the number of those places where the maximum of the random walk
SO,SU... ,Sn is reached, i.e. x(n) is tne largest positive integer for which there
exists a sequence of integers 0 < k\ < k2 < ¦.. < kx(n) < n such that
S(*i) = S(k2) = ¦¦• = S(kx{n)) = M+(n). A3.11)
Csaki (personal communication) evaluated the exact distribution of x(n)- In
he obtained
THEOREM 13.23 For any k = 0,1,2,..., [n/2]; n = 1,2,... we have
P{X(n) = k + 1} = 2-*P{AC_t > *}.
Proof. Consider the sequence
2),X(kx[n)-l + 3),..., X(kx{n))},
where X(l) = Xt = S(l + 1) - 5(/) and A:,, Jt2,... ,kx{n) are defined by A3.11).Let Sj (j = 0,1,2,... ,n
— x(n) + 1) be the sum of the first j of the above
given random variables in the given order. Then {Sy} is a random walk and
x(n) = k + 1 if and only if maxo<j<n-jfc Sj > k which implies the Theorem.
Now we prove a strong law.
NON-ACTIVATEDVERSIONwww.avs4you.com
158 CHAPTER 13
THEOREM 13.24
maxi<jt<nx(fc) 1hm r-= = -
a.s.,n->oo Jg n Z
consequentlyX(n) 1
limsup 7-^—- = - a.s.
n-oo lgn 2
and triviallyx(n) = 1 i.o. a.s.
Proof. Consider the sequence
Then
max y(A;) = max ?.k<n i<M+
ClearlyP{? = k}=2~k {k = 1,2,...; t = 0,1,...)
and the random variables f* (t = 0,1,...) are independent. Hence for any
L = 1,2,... and K = 1,2,... we have
Choosing
K = Kn = ——— lg n and L = Ln =
It
we obtain
PI t^n
if n is big enough. Hence
max ft* 1 _
Hminf — > —-— a.s.n—oo lg n 2
Since M+ > n1/2(lgn)� a.s. (cf. Theorem of Hirsch, Section 5.3) for all but
finitely many n we get
hminf r= — > - a.s.n—oo lgn 2
NON-ACTIVATEDVERSIONwww.avs4you.com
EXCURSIONS 159
Similarly, choosing
K = Kn=]-^lgn and L = Ln = n1/2\gn
we get/ 1 lgn
p{ig«e; > if} = 1-A-^57) «
Let n^T) = >T. Then we have
max ?,* < K a.s.i<L
l ~~
for all but finitely many j where K = Kn.(T) and L = Ln^T). Let jT < N <
{j + 1)T. Then1 +2e
max ?t* < Kn-+1iT) ^ —« lg-W a-s-
\<LN Z
if N is big enough, which in turn implies the Theorem.
LEMMA 13.6 Let
Mk = max{5(pJk), S{pk + 1),..., S{pk+l)} {k = 0,1,... ,n- 1)
and let 0 < M1:n < X2:n < • • • < -Mn.n = M+(pn) be the ordered sample obtained
from the sample Mo, Mi,..., Mn-i- Then for any 0 < e < 1 we have
Mn:n - Mn:n-i > ne a.s.
for all but finitely many n.
Proof. Let
An = An(a,e) =
30 < i < j < n such that r- < Mi, Mj < n{\ogn)a, \M{ - Mj\ < ne \.(logn)a J
Observe that for any i fixed
^— <Mi <n{logn)a\ < P |n
>< MA = ^°*^/
and for any i,j,m with n(logn)~a < m < n(logn)a,0 < i ^ j < n we have
(lognJa\i- MA <ne\Mj = m}<O[ ne
v & '= O
31 -
\ n2
NON-ACTIVATEDVERSIONwww.avs4you.com
160 CHAPTER 13
\\~/- j
^
V nl~e )'Let T be a positive integer with T(l — e) > 1. Then only finitely many of the
events Anr will occur with probability 1. Let nT < N < (n + 1)T. Then
AN Ci4(n+i)TBa,e).Consequently only finitely many of the events An will occur with probability 1.
Since
n(logn)~3 < Mn:n < n(lognK a.s.
(cf. the LIL, the Other LIL and Theorem 11.6) we obtain the Lemma.
Lemma 13.6 and Theorem 13.22 combined imply
THEOREM 13.25 For any C > 0 there exists a D = D(C) > 0 such that
sup ?(M+(n) — j,n) < Dlog3n a.s.
j<Clogn
for all but finitely many n.
In this section as well as in Section 13.3 we investigated the local time of bigvalues. Many efforts were devoted to studying the local time of small values.
Here we mention only the following:
THEOREM 13.26 (Foldes - Puri, 1989). Let
pN = min{k : \Sk\ = N}
and
E{{-ocN, (*N),pn) = H^:0<k<pN, \Sk\ < ocN}.Then for any 0 < a < 1 we have
limsupfc^»MM = 1 „., A3.13)N-oo 2a2iV2loglogiV
and
liminf5lt^^llM2,oglogiV = 1 .... AS.M)
where Co(ct) is the unique root of the equationa
u tan u =
1-a
in the interval @, |].Note that in case a = l,co(a) = tt/2. Hence A3.13) resp. A3.14) are equivalentwith the Other LIL resp. LIL of Khinchine.
NON-ACTIVATEDVERSIONwww.avs4you.com
Chapter 14
A few further results
14.1 On the location of the maximum of a
random walk
Let /x(n) (n = 1,2,...) be the location of the maximum of the absolute value of
a random walk {Sn}, i.e. fi(n) is defined by
M{n) = max \Sk\ = 5(/x(n)) and /x(n) < n. A4.1)
If there is more than one integer satisfying A4.1) then the smallest one will be
considered as /x(n). The characterization of the upper classes of /x(n) is trivial,since /x(n) = n i.o. a.s. In order to get some idea about the lower classes we can
argue as follows.
Since lim^oo /x(n) = oo a.s. by the Law of Iterated Logarithm for any e > 0
(n))| < A + e)Bjx(n) log log ^H
with probability 1 if n is big enough. By the Other LIL for any e > 0
consequently
8 log log n
and
BIT
< A+ eJ/x(n) log log /x(n)
7T2 n
161
NON-ACTIVATEDVERSIONwww.avs4you.com
162 CHAPTER 14
if n is big enough.We ask:
Question 1. Can /x(n) attain the lower bound of A4.2)? The answer is negative.In fact we have
THEOREM 14.1 (Csaki - Foldes - Revesz, 1987).
liminn—>oo
Now we formulate our
Question 2. If
< A "
then by the Law of Iterated
(log log nJn
4 (log log nJ
Logarithm
7T2—
— a.s.
4
for some e > 0 A4.3)
2«) T-,.-," J B log log nI/'/ \ 1/2
. 7T / n \ ,.
2e)-7= :—: . 14.4
We ask: Can | S(n(n)) | attain the upper bound of A4.4) if /x(n) is as small as
possible, i.e. if A4.3) holds? The answer is negative again. In fact we have
THEOREM 14.2 (Csaki - Foldes - Revesz, 1987). Let
Then for any 6 > 0 there exists an e = eF) > 0 such that
2~
A 1/2
_ limsup Mm) < A + 61— a.5.
V n J 2
This theorem roughly says that if /x(n) ~
^fio lognJ 0-e- ^(n) ^s ^^ smaH
possible) then/ \ V2
2 \ log log n
NON-ACTIVATEDVERSIONwww.avs4you.com
A FEW FURTHER RESULTS 163
Question 3. Intuitively it is clear that M(n) can be (and will be) small if /x(n)is small. Theorem 14.2 somewhat contradicts this feeling. It says that if /x(n) is
as small as possible then M(n) will be small but not as small as possible without
having any condition about /x(n). It will be f (j^^I/2 instead of -^which is the smallest possible value of M(n) by the Other LIL. We ask: How
small can /x(n) be, if M[n) is as small as possible? The answer is:
THEOREM 14.3 (Csaki - Foldes - Revesz, 1987). For any L > 0 there exists
an e = e{L) > 0 such that with probability 1 the inequalities
and
cannot occur simultaneously if n is big enough. However, if g(n) is a positivefunction with g(n) / oo then for almost all u> 6 ft and e > 0 there exists a
sequence 0 < nx = n1(a;,e) < n2 = n2(u;,e) < ... such that
V2
*< g{nk)T. ; rr and hm M(nk) =
-p.'(loglognjkJ »-°°V nk J /8
Question 4. Instead of Question 3 one can ask: How big can /x(n) be, if M(n)is as small as possible? The answer to this question is unknown.
The following theorem gives a joint generalization of the above three theo-
theorems. It also contains the LIL and the Other LIL (cf. Sections 4.4 and 5.3).THEOREM 14.4 (Csaki - Foldes - Revesz, 1987). Let
(log log nJain) = mW,
71
1/2
Then the set of limit points of the sequence (a(n),b(n)) as (n —> oo) is K with
probability 1.
NON-ACTIVATEDVERSIONwww.avs4you.com
164 CHAPTER 14
Remark 1. This theorem clearly does NOT imply that (a(n),b(n)) 6 K or even
(a(n),b(n)) belongs to a neighbourhood of K if n is big enough. However,
{a(n), b(n)) belongs to a somewhat larger set Ke D K if n is big enough. In fact
we have
THEOREM 14.5 (Csaki - Foldes - Revesz, 1987). Let
Ke = {(*,y) : x > 0,y > 0, ^ + ~ < 1 + e| (e > 0).
T/icn for any e > 0
(a(n),6(n)) €/fg a.s.
/or a// 6uf finitely many n.
In order to formulate a simple consequence of Theorem 14.4 let R*(n) be the
length of the longest flat interval of {M(k),0 < k < n}, i.e. R*(n) is the largestpositive number for which there exists a positive integer a such that
0 < a < a + R* (n) < n
and
Af(a) =M{ot + R*{n)).Then by Theorem 14.1 (or 14.4) we have
THEOREM 14.6 (Csaki - Foldes - Revesz, 1987).
. (log log nJ 7T2hminf In — K (nil = —
a.s.n-oo n 4
Equivalently for any e > 0
, 7T2 n
7T2 n
As far as the lower classes of R* (n) are concerned we have
THEOREM 14.7 (Csaki - Foldes - Revesz, 1987).
lim.nfloglognn-oo n
'
NON-ACTIVATEDVERSIONwww.avs4you.com
A FEW FURTHER RESULTS 165
where 0 is the root of the equationoo okok
= 1
(cf. Theorem 13.3). Equivalently for anys > 0
and
Remark 2. In Theorems 13.1 and 13.3 we investigated the length of the longestflat interval of M+(n). Comparing our results regarding the upper classes we
obtain the intuitively clear fact that the longest flat interval of M+(n) can be
(and will be) longer than that of M(n). Comparing the known results regardingthe lower classes no difference can be obtained.
About the proofs of the above theorems we mention that they are based on
the following:
THEOREM 14.8 (Imhof, 1984). Let ut{x,y) be the joint density of (r^),t~ll2rn{t)) where M(t) is the location of the maximum of a Wiener process. Then
y
14.2 On the location of the last zero
Let ^(n) be the location of the last zero of a random walk {5jk,A: < n}, i.e.
?(n) = max{A: : 0 < k < n, Sk = 0}.
Theorem 13.1 claims that: if g(n) is a nondecreasing sequence of positive numbers
then
if and only if
n=l
NON-ACTIVATEDVERSIONwww.avs4you.com
166 CHAPTER 14
Consequently for any e > 0
)) and 7T7TI^LUC(tf(n)).(lognJ+e
Since ^(n) = n i.o.a.s. and ^(n) < n the description of the upper classes of
^(n) is trivial.
Here we wish to investigate the properties of the sequence {^(n)} for those
n's only for which Sn is very big or M(n) is very small. It looks very likely that
if Sn is very big (e.g. Sn > BnloglognI/2) then ^(n) is very small. In the
next theorems it turns out that this conjecture is not exactly true. In order to
formulate our results introduce the following notations.
Let f(n) = n1/2jf(n) G ULCEn) with g(n) f oo. Define the infinite random
set of integersZ = Z(f)={n:Sn>f(n)}.
Furthermore, let a(n),/?(n) be sequences of positive numbers satisfying the fol-
following conditions:
a. (n)nonincreasing,
0 < a{n) < 1,
P{n) I 0,
nct(n) | oo, n/3(n) | oo.
Then we have
THEOREM 14.9 (Csaki - Grill, 1988).
na[n) <EUUC(#(n),n€ 2)
if and only if
Further,
nC(n) E LLC(#(n),n € 2)
i/ and only if
oo -I / 2 fn\ \
^(o'jfi') = yz ~02(n)/^2(n) exp (—I < °°-
^in V 2 /
NON-ACTIVATEDVERSIONwww.avs4you.com
A FEW FURTHER RESULTS 167
Remark 1. na{n) 6 UUC(#(n),n 6 2) means that na(n) > V(n) a.s. for
all but finitely many such n for which n ? Z. In other words the inequalitiesSn > f{n) and #(n) > na(n) simultaneously hold with probability 1 only for
finitely many n.
In order to illuminate the meaning of the above theorem we present two
examples.
Example 1. Let f(n) = (B - e)n log log nI/2^ < e < 2). Then we obtain that
the inequalities
Sn> (B-e)nloglognI/2 and #(n)>J(l + e)n
hold with probability 1 only for finitely many n. However,
Sn > (B - e)nloglognI/2 and V{n)>Ul- e)n i.o. a.s.
The above two statements also follow from Strassen's theorem (Section 8.1).Further,
Sn>{{2- e)nloglognI/2 and #(n) < n(logn)"n i.o. a.s.
if and only if 77 < e. Note the surprising fact that ^f(n) < n(logn)� i.o.
a.s. but there are only finitely many n for which ^(n) < n(logn)� and Sn >
(InloglognI/2 (say) simultaneously hold.
Example 2. Let f(n) = BnloglognI/2. Then we obtain that for any e > 0
the inequalities
Sn > BnloglognI/2 and tf(n) > (- + e) }^^n\2 / log2n
hold with probability 1 only for finitely many n. However,
Sn > BnloglognI/2 and ^(n) > --^^ i.o. a.s.2 log2 n
Further,
Sn > BnloglognI/2 and ^(n) < n(loglogn)"f? i.o. a.s.
if and only if 77 < 4.
Now we turn to our second question, i.e.. we intend to study the behaviour of
^ for those n's for which M{n) is very small (nearly equal to 7m1/2 (8 log log n)�/2,
NON-ACTIVATEDVERSIONwww.avs4you.com
168 CHAPTER 14
cf. the Other LIL E.9)). In this case we can expect that ty(n) is not very small.
The next theorem shows that this feeling is true in some sense. In order to
formulate it introduce the following notations. Let "/(n) and 6(n) be sequences
of positive numbers satisfying the following conditions:
0<-y(n),*(n) < 1,
6(n) nonincreasing, 6(n)n1/2 f oo,
"/(n) monotone, ^{n)n | oo,
))~2 monotone,
o[n)
Then we have
THEOREM 14.10 (Grill, 1987/A).
P{#(n) < ni{n),M(n) < 6{n)n1'2 i.o.} = 0 or 1
depending on whether 73('y,5) < oo or I^jS) = oo where
Consequence. The limit points of the sequence
are the set
{(x,y) : y2 > 4 - 3x,0 < x < 1}.
Example 3. The inequalities
and
/ n \1/2
hold with probability 1 only for finitely many n, i.e. if
/ \ 1/2
then Mln) > D - 3-/ - eW2 —— a.s.
NON-ACTIVATEDVERSIONwww.avs4you.com
A FEW FURTHER RESULTS 169
for all but finitely many n. Similarly if
/ \ 1/2
M(n) < D-37-eI/27r -— then tf(n) > -yri a.s.
\81oglogny
for all but finitely many n. This means that if M(n) is very small then ^(n)cannot be too small. For example, choosing D - 3"/ - eI/2 = 1 + 6 we have: if
/ \ 1/2
M(n) <(l+6)ir\ then tf(n) > A - 6)n a.s.v ; ~ v ;
\81oglogn;~
for all but finitely many n.
Having the above result we formulate the following conjecture:For any 6 > 0 and for almost all w € ft there exists a sequence of integers
0 < nx = ni(u;,?) < n2 = ri2(u},6) < ... such that
/ \ 1/2
Iand *(nO = nt (t = l,2,...).O<(l + tyr\ 8 log log rii)
The proofs of the above two theorems of this paragraph are based on the eval-
evaluation of the joint distribution of ^(n) and Sn. Here we present such a result
formulated to Wiener process.
THEOREM 14.11 (Csaki - Grill, 1988). Let x > 0,0 < y < 1. Then
> ty,W(t) > xt1'2} = ^,y»;, exp \-where ip(t) is the last zero of W(s),0 < s <t, i.e.
ip{t) = sup{s : 0 < s < t,W(s) = 0}.
2A -y)
14.3 The Ornstein - Uhlenbeck process and a
theorem of Darling and Erdos
Consider the Gaussian process {V(t) = t~ll2W{t)\Q < t < oo}. Then EF(t) =
0, EF2(t) = 1 and EF(t)F(s) = \fs~ft,s < t. The form of this covariance
function immediately suggests that, in order to get a stationary Gaussian process
out of V(t), we should consider
Ua(t) = V(eat), -oo<t<+oo {a fixed > 0).
This latter process is a stationary Gaussian process, for 1&Ua(t)Ua(s) = e~a^~^'2,and it is called Orstein - Uhlenbeck process. We will use the notation U(t) =
f/2(t), and mention, without proof, the following:
NON-ACTIVATEDVERSIONwww.avs4you.com
170 CHAPTER 14
THEOREM 14.12 (Darling- Erdos, 1956).
lim P{ sup U(t) < a{y,T)} = exp(-e-tf), A4.5)T—oo 0<t<T
lim P{ sup \U(t)\ < a{y,T) = exp{-2e~v), A4.6)T—>oo 0<t<T
where for any —oo < y < oo
a{y,T) = (y + 21ogT + i log log T - ^ log*) BlogT)-1/2.
We also mention a large deviation type theorem of Quails and Watanabe A972)(cf. also Bickel - Rosenblatt, 1973).
THEOREM 14.13 For any T > 0 we have
'/''
= Lz-»oo J 0<KT
Applying their invariance-principle method Darling and Erdos A956) also proved
THEOREM 14.14
and
Jjrn P{max AT^S* < a{y, logn)} = exp(-e-tf)
lim P{max Ar~1/2 | Sk |< a(y,logn)} = exp(-2c-tf)n—> oo 1 < Jk < n
for any -oo < y < oo.
A strong characterization of the behaviour of maxi<jk<n k~l/2Sk is given in the
following:
THEOREM 14.15
max1<Jk<nA: Sk,lim —;
^-=
rrjz—= 1 CIS.
n-oo B log log nI/2
Proof. The LIL of Khinchine implies that
maxi<Jk<n k lSklimsup—;—-=-^r tttt— < 1 a.s.
n-oo Bl0gl0gTlI/2-
NON-ACTIVATEDVERSIONwww.avs4you.com
A FEW FURTHER RESULTS 171
Applying Theorem 5.3 we obtain that for any n big enough with probability 1
there exists a
with
such that
Hence
> K1-
max -^ > —?= > \l2\og\ogK, > v^loglogn1"*" > (l - e) \J2 log log nl<*<n y/k y/K
V
and we have Theorem 14.15.
It looks also interesting to study the limit behaviour of the sequence
Ln = max max AT1/2(Si+Jk - S,) (n = 1,2,...).0<j<n l<Jfc<n-j
We prove
THEOREM 14.16
l<liminf——\r-jz < limsup 7——\rjz = K < 00 a.s.n->0° BlognI/2 n_>0O BlognI/2
where the exact value of K is unknown.
Proof. Let an = [(log n)a] (a > l). Then by Theorem 7.13 (see also Section 7.3)for any e > 0 we have
Ln> B logn)�/^ max a^S,-^ - 5,)) > 1 - e a.s.
B log n) 1/2
which proves the lower part of the Theorem.
In order to prove its upper part the following result of Hanson and Russo
A983/A E.11 a)) will be utilized:
If an = f(n) logn with f(n) / 00 then
limsup sup sup , }—> ,,
} < 1 a.s.
n—00 0<j<nan<k<n-j B1ognj1/^
NON-ACTIVATEDVERSIONwww.avs4you.com
hmsup sup sup ,= < 1 a.s.
n—oo 0<j<n21ogn<Jk<an 2alOgn
hmsup sup sup ,= < 1 a.s.
n-»oo 0<j<n l<Jk<21ogn /2k lOgU
172 CHAPTER 14
Applying again Theorem 7.13 we obtain
k-lt*{Si+k - Sj) ^
hmsup sup sup j====—<
n—oo 0<j<n21ogn<fc<an W2/(n) log n
and clearly
Since f(n) may converge to infinity arbitrarily slowly we obtain the Theorem
by the Zero-One Law.
We present on the value of K of Theorem 14.16 the following:
Conjecture. K = 1.
Let u{n) = u(n,S) resp. u(T) = u(T,W) be the smallest integer resp. the
smallest positive real number for which
resp.
It looks an interesting question to characterize the properties of v(n,S) resp.
u(T,W). Clearly
u(n,S)=n resp. u{T,W) = T i.o. a.s.
On LLC(i/(-,-)) we have
THEOREM 14.17 For any e > 0
exp({\ognI-') €LLC{u{n,-)).
Proof. By Theorem 14.15 and the LIL of Khinchine we have
which implies Theorem 14.17.
Remark 1. Since the Invariance Principle (cf. Section 6.2) only implies that
max Jfc�/25jk - max r1/2W(*)| < Oil) a.s.,l<*<n l<t<n
\ j\ - \ j
NON-ACTIVATEDVERSIONwww.avs4you.com
A FEW FURTHER RESULTS 173
we cannot get Theorem 14.14 from Theorem 14.12 by the Invariance Principle.However, applying Theorem 14.17 we obtain
max k~l'2Sk - max rl^W(t)\ < O (expHlognI"8)) a.s.Kk<n
for any e > 0. Hence we obtain Theorem 14.14 via Theorem 14.12 and the
Invariance Principle.Studying the strong behaviour of U(t) Quails and Watanabe A972) proved
THEOREM 14.18 For any e > 0
and
14.4 A discrete version of the Ito formula
Ito A942) defined and studied the so-called Itd-integral
f f(W(s))dW(s)Jo
where /(•) is a continuously differentiate function. Here we do not give the
definition but we mention an important property of this integral, the celebrated
ITO-FORMULA (Ito, 1942).
I f(W(s))dW(s) = / f(X)d\ - I ^JO Jo Jo
-ds. A4.7)
In fact A4.7) is a special case of the so-called Ito-formula. Here we are interested
to find the analogue of A4.7) for random walk. In fact we prove
THEOREM 14.19 (Szabados, 1989). Let f{k)(k € Z1) be an arbitrary func-function and define
g(k) = 0
if *>x,
if k = 0,
j=k+l
NON-ACTIVATEDVERSIONwww.avs4you.com
174 CHAPTER 14
Then for any n = 0,1,2,... we have
i=0 t=0
Remark 1. The function g(-) can be considered as the discrete analogue of the
integral /on f{X)d\, X-+\{f{Si+i) - f{Si)) is the natural discrete version of /'($)and YX=i f{Si)Xi+i can be considered as the discrete Itd-integral.
Proof of Theorem 14.19. We get
(In order to check A4.9) consider the six cases corresponding to 5, = 0, S; >
0,Si < 0;Xi+i = l,Xi-i = -1 separately.) Summing up A4.9) from 0 to n we
obtain A4.8).
Example 1. Let f(t) = t. Then by A4.7) we have
W{S)dW{S) = V^ft-t- A4.10)
and by A4.8)n
ri 4- 1 ^^ «
SiXi+l = g{Z>n+1)— = —
-. A4.HJi=0
ILL
A4.10) and A4.11) completely agree.
Example 2. Let f(t) = i1. Then by A4.7) we have
[tW2(s)dW{s) = ^-^- - ftW{s)ds A4.12)Jo 3 Jo
and by A4.8)
E SfXi+l = g(Sn+l) - E S< - %i = % _ ? Si _ ^ti. A4.13)t=0 t=0
* 6t=0
The term —5n+1/3 of A4.13) is not expected by A4.12). However, we know that
the orders of magnitude of the terms E?=o s<Xi+i, 5^+1 /3 and ?t?=o S{ are n3/2,while that of 5n+i is n1/2 only.
Remark 2. Applying the Invariance Principle 1 A4.7) can be obtained as a
consequence of A4.8).
NON-ACTIVATEDVERSIONwww.avs4you.com
A FEW FURTHER RESULTS 175
The celebrated Tanaka formula gives a representation of the local time of a
Wiener process via It6-integral.
TANAKA FORMULA (cf. McKean, 1969). For any x 6 R1 and t > 0 we
have
rj{x,t) = \W(t) -x\-\x\- fts\gn{W{s)-x)dW(s).Jo
Here we are interested in giving the discrete analogue of this formula. In order to
do so instead of f (•, •) we consider a slightly modified version of the local time.
Let
?*(z,n) = #{A: : 0 < k < n,Sk = x).
Then we have
THEOREM 14.20 (Csorgo - Revesz, 1985). For any x € Z1 and n = 1,2,...
n-l
?(x,n) = \Sn -x\-\x\-Y, signEfc - x)Xk+l. A4.14)Jk=O
Proof. Observe that
I — 1 if x = 0,
signEfc -x)Xk+i = -1 (i = l,2,...),
signEfc - x)Xk+i = \Sn - x\ - 1
where
fi=iC{x,n) if x^O,
The above three equations easily imply A4.14).
Remark 3. The Tanaka formula can be proved from A4.14) using Invariance
Principle 1.
Jk=O
n-l
NON-ACTIVATEDVERSIONwww.avs4you.com
Chapter 15
Summary of Part I
Exact
distr.
Limit
distr.
Upperclasses
Lower
classes
Strassen
type theorems
Sn Th. 2.1 Th.'s
2.9, 2.10
EFKP LIL
Sect. 5.2
Th. 5.1 Strassen's
Th. 1. Sect. 8.1
Mn Th. 2.6 Th. 2.13 EFKP LIL
Sect. 5.2
Th. of
ChungSect. 5.3
Th. 8.2;Wichura's
Theorem
Sect. 8.4
Th. 2.4 Th. 2.12 EFKP LIL
Sect. 5.2
Th. of
Hirsch
Sect. 5.3
Th. 8.2
Th. 9.3 (9.11) Th. 11.1 Th. 11.1 Th. 11.16
The limit behaviour of f@,n) is the same as
that of M+ by Th. 10.3
Th. 9.3 | (9.9) [ Th. 11.6 | Th. 11.6 | Th. 11.17
Since f @,pn) = n a description of f@,n) gives a
description of pn
Z{x,n) Th. 9.4 The limit behaviour is
the same as that of
f@,n) for fixed x,n —> oo.
Th. of
Donsker-
Varadhan
Sect. 11.4
Th. 9.5 The limit behaviour is the same as that of
f@,n)/2 (cf. A0.14)) or M+/2 (cf. Th. 10.4).
177
NON-ACTIVATEDVERSIONwww.avs4you.com
178 CHAPTER 15
*W
/x(n)
X(n)
Exact
distr.
Th. 9.8
Th. 9.9
Th. 9.10
Th. 13.23
Th. 2.7
Limit
distr.
Th. 9.14, A0.6)(9.12)(9.12)
The limit
tl
(9.12)
Th. 2.14
Upperclasses
Th. 11.5
Th. 13.1
Trivial
Th. 13.1
Th. 13.1
sehaviour ol
lat of R{n)Th. 14.6
Trivial
Trivial
Th. 13.24
Th. 7.2
Lower
classes
Th. 11.4
Th. 13.2
Th. 13.1
Th. 13.3
Th. 13.3'
R{n) is the
by Th. 10.3
Th. 14.7
Th. 14.1
Th. 13.1
Th. 13.24
Th. 7.3
Strassen
type theorems
same as
Replacing Sn, Mn,M+, ?@, n)... by W(t),m(t),m+(t),rj@, t),... respectively the
above-mentioned results remain true by the Invariance Principle 1 (cf. Section
6.3) and Theorem 10.1, with the exception that there is no immediate analogueof 0(n) and the natural analogue of x{n) does n°t have any interest.
In some cases we also investigated the joint behaviour of the r.v.'s of the
above table. A table for these results is
Mn
AC
ACTh.'s 2.5, 2.6,5.8
M~
Th.'s 2.5, 2.6,5.8
Th.'s 5.5, 5.6
Th.'s 14.4, 14.5
14.8
tf(n)Th. 14.9
Th.'s 14.10, 14.11
Clearly many of the results of Part I are not included in the above two tables.
For example, the results about increments, the rate of convergence of Strassen-
type theorems, the results on the stability of the local time, etc. are missing from
the above tables. A summary on the increments of the Wiener process is givenat the end of Section 7.2.
NON-ACTIVATEDVERSIONwww.avs4you.com
"The earth was without form and void,and darkness was upon the face of the
deep."
The First Book of Moses
II. SIMPLE SYMMETRIC
RANDOM WALK IN ZdNON-ACTIVATEDVERSION
www.avs4you.com
Notations
1. Consider a random walk on the lattice Zd. This means that if the movingparticle is in x ? Zd in the moment n then at the moment n + 1 the
particle can move with equal probabilities to any of the 2d neighbours of
x independently of how the particle achieved x. (The neighbours of an
x ? Zd are those elements of Zd whose d — 1 coordinates coincide with
the corresponding coordinates of x and one coordinate differs by +1 or — 1
from the corresponding coordinate of x.)Let Sn = S(n) be the location of the particle after n steps (i.e. in the
momemt n) and assume that So = 0. Equivalently: Sn = X\ + X2 + • • • +
Xn(n = 1,2,...) where Xx, X2,... is a sequence of independent, identicallydistributed random vectors with
= e,} = V{XX = -e,} = ^ (i = 1,2,... d)
where t\,t2,...,td are the orthogonal unit-vectors of Zd.
d
2. For any x = (xu x2,..., xd) e Rd let ||i||2 = ?>,2.
3. Mn = M(n)= max ||5,||.
4. ?(x, n) = #{fc : 0 < k < n, Sk = x} (x 6 Zd, n = 1,2,...).
5. f(n) = maxf(z,n).
6. px = min{/c : k > 0, Sk = 0},pi = min{/c: k> pi,Sk = 0},
pn = m\n{k : k > pn-i,Sk = 0}.
7. p2k = P{S2k = 0}, p2k+1 =0 (k = 0,1,-2,...).
181
NON-ACTIVATEDVERSIONwww.avs4you.com
182 II. SIMPLE SYMMETRIC RANDOM WALK IN Zd
8. q2k='P{S2k=0,S2k-2 ^0,...,54^0,52^0}=P{p1 = 2k} =
P{?@,2*)=l, f@,2A:-l)=0} (A: = 1,2,...)-
9- In = P{52 ± 0,54 ± 0,... ,S2Jk_2 ^ 0} = P{e@,2A: - 2) = 0} =
fan (fc = 2,3,...) and i2 = 1.
10. -y = Jlm-y^ = P{nlim f@,n) = 0} = P{Pl = oo} = 1 -
oo n-oo,= i
11. Let W(t) = {Wl{tIW2{tI...1Wd{t)IvrhereWl{tIWt(tI...1Wd[t) are in-
independent Wiener processes. Then the Rd valued process W(t) is called a
d-dimensional Wiener process.
12. m{t) =
NON-ACTIVATEDVERSIONwww.avs4you.com
Chapter 16
Recurrence theorem
This chapter is devoted to proving the
RECURRENCE THEOREM (Polya, 1921).
PI9 =0¦ \=l1 if d~2'
*¦ n*" '*
[0 if d > 2.
This Theorem was proved for d = 1 in Section 3.1. Hence we concentrate on
the case d > 2.
LEMMA 16.1 For any n = 1,2,...; d = 1,2,...
B»)!= 0} =
nd)
Proof is trivial by a combinatorial argument.
LEMMA 16.2 For any d= 1,2,... as n —> oo we have
( d \d/2P{52n = 0}.« 2 (_j .
Further, in case d = 2
183
NON-ACTIVATEDVERSIONwww.avs4you.com
184 CHAPTER 16
Proof can be obtained by the Stirling formula.
For later reference we give the following analogues of Lemmas 16.1 and 16.2.
LEMMA 16.3 Let d = 2. Then
2n 2n-k
keAn(x,y)
provided that
where
x + y= 0 (mod 2) and \x\ + \y\ < 2n
keAn(x,y) if and only if k = x (mod 2) and \x\ < k < 2n - \y\.
Proof is trivial by a combinatorial argument.
LEMMA 16.4 (Erdos - Taylor, 1960/A, B.9) and B.10)). Let d = 2,x = (xi,x2) and xx + i2 = 0 (mod 2). Then
P{S2n = x}7TM
if n>\\x\\\
<(-L+O(n-2))exp(-H!) if n<
Proof can be obtained by the Stirling formula.
Similarly one can obtain
LEMMA 16.5 Let d > 2, x = (xx,x2,.. .,xd) and xx + x2 + ••• + xd
(mod 2). Then
= 0
P{S2n =
LEMMA 16.6 In case d = 2
2n
if n>\\x\\\
if n<\\x\\\
lim — = 1.
k=l
NON-ACTIVATEDVERSIONwww.avs4you.com
RECURRENCE THEOREM 185
Proof. Clearly
E E p{^i = 0, S2k = 0} = 2 ?2 P{S2j = 0,S2j+2fc = 0} + ? p{52i = 0}
= 2
2 ^
Hence we have Lemma 16.6.
Proof 1 of the Recurrence Theorem. In the case d = 2 our statement follows
from Lemmas 16.2, 16.6 and Borel - Cantelli lemma 2* of Section 4.1, while in
the case d > 3 it follows from Lemma 16.2 and Borel - Cantelli lemma 1 (cf.Section 4.1).
Remark 1. Lemma 16.6 is also true in the case d = 1. (The proof is essentiallythe same.) Hence we obtain a new proof of the recurrence theorem in the case
d = 1. The third proof (cf. Section 3.2) applied in case d — 1 does not work in
the case d > 2. The idea of the first proof can be applied in the case d = 2 but
it requires hard work. The second proof can be used without any difficulty.
Proof 2 of the Recurrence Theorem. Introduce the following notations:
Po = l, P2k = P{S2k =
O}*2y4k^j ,
q2k = P{S2k = 0,52Jk_2 ^ 0,52Jk_4 ^ 0,..., S2 ^ 0},
P(z) = f>2fcz2\ Q{z) = fjq2kz2k.
Between the sequences {p2jt} and {<72jt} one can easily see the following relations:
@) po = 1,
A) p2 = q2,
(ii) p\ = <74 "H qiPii
(iii) p6 = q6 + q4p2-
(k) p2jfc = <72Jfc + <72Jfc-2P2 H h <72P2Jfc-2>
NON-ACTIVATEDVERSIONwww.avs4you.com
186 CHAPTER 16
Multiplying the k-th equation by z2k and adding them up to infinity we obtain
P(z)=P{z)Q{z) + l i.e. Q(z) = l-J^. A6.1)
By Lemma 16.2 we obtain
hm P(z) {= oo if d < 2,< oo if <i > 3,
i.e.
if d < 2,if <i>3.
Since Q(l) = Yl<?=iQ2k is the probability that the particle eventually visits the
origin we obtain
oo \~l
< 1 if d > 3,
"=1
[I if d = 2.
Hence we have the theorem.
Remark 2. In the case d > 3 the formula
P{5n = 0 for some n = 1,2...} = 1 - P{ lim f@, n) = 0}
l-l^— A6.2)
Jk=O
is applicable to the evaluation of the probability that the particle returns to the
origin. In fact by Lemma 16.2 we have
Jk=O Jk=n Jk=O
where Y%ZoP2k can be numerically evaluated by Lemma 16.1. For example, in
the case d = 3 one can obtain Z)*Li 92* ~ 0.35. In fact this method is not
easily applicable for concrete calculations. Griffin A989) gave a better version
of it and evaluated the probability of recurrence for many values of d. For
example, for d = 3,4,20 he calculated that the probabilities of recurrence are
NON-ACTIVATEDVERSIONwww.avs4you.com
RECURRENCE THEOREM 187
0.340537, 0.193202, 0.026376. Note that in the case d = 20, P{52 = 0} = 0.025.
Hence the probability that the particle returns to the origin but not in the second
step is 0.001376.
It looks also- interesting to estimate the probability
n-l
that the particle returns to the origin but not in the first 2n - 2 steps. We prove
LEMMA 16.7 (Dvoretzky - Erdos, 1950). For any d > 3
>2n ~ 2) = °> ~
1
A6.3)k=rt
Proof. Classifying the paths according to the last return to the origin, we get
n
2.= °'5i 5* °'.i = 2i + l> + 2,... ,2n}
t=0
n
EPE2. = °'5i " S* * °5i = 2i + l>2i + 2,... ,2n}i=0
n
Subtractingn
EP2.^« = ^— A6.5)
i=0
from both sides of {16.4) we have
A6-6)
Since q2n \ 1 we have
n
1 - An > l2n+2- An + -y J2 Pn- (I6-7)
NON-ACTIVATEDVERSIONwww.avs4you.com
188 CHAPTER 16
Since
)t=l \t=0 /
(cf. 10 of Notations and A6.2)) we obtain (by A6.5), A6.8), A6.7) and 9 of
Notations)
P*i = l~An> l2n+2~ An +
t=0 t'=0
n n
= l2n+2 -lJ2PK+iY2 PK = "t**+* ~^=t=0 t=l j=n+l
Consequently (by A6.8))
. A6-9)
t=0
Hence we have A6.3) by Lemma 16.2.NON-ACTIVATEDVERSIONwww.avs4you.com
Chapter 17
Wiener process and Invariance
Principle
Let Wx(t), W2(t),..., Wd{i) be a sequence of independent Wiener processes. The
Rd valued process W(t) = {Wx(t),W2{i),. ..,Wd{i)} is called a d-dimensional
Wiener process. We ask how the random walk Sn can be approximated by H^(t).The situation is very simple if d = 2.
Consider a new coordinate system with axes y = x, y = — x. In this coordinate
system
Xn = A,0) in the original system ,
Xn = @,1) in the original system ,
xn = (-1,0) in the original system ,
xn = @,-1) in the original system .
Observe that the coordinates of Xn are independent r.v.'s in the new coordinate
system (it is not so in the old one); hence by Invariance Principle 1 of Section
6.3 there exist two independent Wiener processes W\{t) and W2(t) such that
'
B-V2,(-2�/(-2�/
k B-1'2,
2-1/2)2? 2-1/2)2?_2-l/2)_2-i/2)
if
if
if
if
\2-1/2Wx (n) - S™ | = O(log n) a.s.
and
a.s.
where SW,S$*) are the independent coordinates of Sn in the new coordinate
system. Consequently we have
THEOREM 17.1 Let d = 2. Then on a rich enough probability space {fl, 7,P}one can define a Wiener process W(-) ? R2 and a random walk Sn € Z2 such
that.i _ i/•> i \ii —/• \
a.s.
189
NON-ACTIVATEDVERSIONwww.avs4you.com
190 CHAPTER 17
In the case d > 2 the above idea does not work. Instead we define
/C?) = #{k : 1 < k < n,Xk = e, or - e,} (t = 1,2,... ,d)
where e< is the t'-th unit vector in Zd. Then by the LIL we have
a.s. A7.1)
for any e > 0 and for all but finitely many n.
Let Sn = (SJil\SJ?\...,SJ4) (where SW is the i-th coordinate of Sn in the
original coordinate system). Then by Invariance Principle 1 there exist indepen-independent Wiener processes WX(-),W2{-),...,Wd{-) such that
= O(log /CW) = O(logn) a.s.
for any i = 1,2,..., d. By A7.1) and Theorem 7.13 we have
a.s.
Consequently we have
THEOREM 17.2 On a rich enough probability space {fi,7,P} one can definea Wiener process W(-) ? Rd and a random walk Sn 6 Zd such that
~ W il logn
for any d = 1,2,
It is not hard to prove that
P{W(t) = 0 i.o.} =1 if d = 1
and
P{W(t) = 0 for any t > 0} = 0 if d > 2.
Hence we can say that the Wiener process is not recurrent if d > 2. However, it
turns out that it is recurrent in a weaker sense if d = 2.
THEOREM 17.3 (see e.g. Knight, 1981, Th. 4.3.8). For any
e > 0 we have
P{\\W(t)\\<e t.o.} = l •/ d = 2, A7.2)
P{|lwr(*)ll < e- i.o.} = 0 if d>Z A7.3)where i.o. means that there exists a random sequence 0 < t\ = ti(u>,e) < t2 =
t2{u,e) < ... such that limn_oo tn = oo a.s. and ||W(tn)|| < e (n = 1,2,...).
NON-ACTIVATEDVERSIONwww.avs4you.com
WIENER PROCESS AND INVARIANCE PRINCIPLE 191
The proof of Theorem 17.3 is very simple having the following deep lemma
which is the analogue of Lemma 3.1.
LEMMA 17.1 (Knight, 1981, Th. 4.3.8). Let 0 < a < b < c < oo. Then
P{inf{s : s > 0, \\W(t + s)\\ = a} < inf{s : s > 0, \\W{t + s)\\ = c} \ B}
' c-b
c — a
logc-
logc -
c*-d-
log 6
log a
b2-<t
if
if
if
d =
Au —
d>
1,
2,
3,(> ci~a - a'
where B = {\\W{t)\\ =6}.
Remark 1. A7.3) is equivalent to
lim ||W(i)|| = oo a.s. if d > 3. A7.4)t-»oo
The rate of convergence in A7.4) will be studied in Chapter 18.
In connection with A7.2) it is natural to ask how the set of those functions
et can be characterized for which
T{\\W{t)\\<et i.o.} = l. A7.5)
This question was studied by Spitzer A958), who proved
THEOREM 17.4 Let g(t) be a positive nonincreasing function. Then
g(t)t^eLLC(\\W(t)\\) (d = 2)
if and only if ?~=1(fc | log </(*:) I)� < oo.
Remark 2. Theorem 17.4 implies
(log0. fLLC(||W(t)||) if e>0,1 e
\)) if ?<0.
The proof of Theorem 17.4 is based on the following:
NON-ACTIVATEDVERSIONwww.avs4you.com
192 CHAPTER 17
LEMMA 17.2 (Spitzer, 1958). For any 0 < tx < t2 < oo we have
Here we also mention a simple consequence of Theorems 2.12 and 2.13 (cf.also Theorem 6.3).
THEOREM 17.5 For any d= 1,2,... and T > 0 we have
P(m(T) > uT1/2) = O(trVu3/2) as u -> oo
and
P(m(T) < uT1'2) = exp(-O(u-2)) as u -> 0.
Similarly for any d= 1,2,... as N —> oo we have
?{M{N) > uN1'2) = exp(-O(u2)) if u -> oo but u < Nx'z
and
Y{M{N) < uNxl2) = exp(-O(u�)) if u -> 0 6u* u > iV�/3.NON-ACTIVATEDVERSION
www.avs4you.com
Chapter 18
The Law of Iterated Logarithm
At first we present the analogue of the LIL of Khinchine of Section 4.4.
THEOREM 18.1
\imsnpbt\\W{t)\\ = l a.s. A8.1)t-»oo
and
limsup6n<i1/2||Sn|| = 1 a.s. A8.2)n—»oo
where bt = B* log log*)�/2.
Proof. By the LIL of Khinchine we obtain
> 1.*¦ -
rr \ / r r—
t-»oo
In order to obtain the upper estimate assume that there exists an e > 0 such
that
limsupbt\\W{t)|| > 1 + e: a.s.
t-»oo
(Zero-One Law (cf. Section 3.2)). For the sake of simplicity let d = 2 and define
4>k = karccos0,(k = 0,1,2,... ^(arccos©)�]) where 0 = A + er/2)(l + er)�.If 6t||VV(t)|| > 1 + e then there exists a k such that
bt\cos<j>kWx(t) +sxn<{>kW2{t)\ > 1 + |. A8.3)
Since cos 4>kWi{i) + sin<f>kWi(t){t > 0) is a Wiener process A8.3) cannot occur if
t is big enough. Hence we have A8.1) in the case d = 2. The proof of A8.1) for
d > 3 is essentially the same. A8.2) follows from A8.1) by Theorem 17.2.
Applying the method of proof of Strassen's theorem 2 of Section 8.1 we obtain
the following stronger theorem:
193
NON-ACTIVATEDVERSIONwww.avs4you.com
194 CHAPTER 18
THEOREM 18.2 The process {btW(t),t > 0} is relatively compact in Rd with
probability 1 and the set of its limit points is
Cd = {xeRd,\\x\\<\}.
The real analogue of Strassen's theorem can also be easily proved. It goes
like this:
THEOREM 18.3 The net {btW(xt),0 < x < 1} is relatively compact in
C@,1) x • • • x C@,1) = (C@, l))d and the set of its limit points is S* where Sd con-
consists of those and only those Rd valued functions f(x) = (fi{x), fi{x),..., fd{x))for which /,@) = 0,/,-(•)(i = 1,2...,d) is absolutely continuous in [0,1] and
We ask about the analogue of the EFKP LIL of Section 5.2. It is trivial to
see that if a(t) ? ULC{H^(t)} in the case d = 'l then the same is true for any d.
However, the analogue statement for UUC{Wr(t)} is not true. As an example,we mention that Consequence 1 of Section 5.2 tells us that in the case d = 1 for
any e > 0
Sn < [2n Hog log n + ( - + e) log log log n) J1/2
a.s.
for all but finitely many n. However, it turns out that in case d > 1 it is not
true. In fact for any d > 1
l l l ) J i.o. a.s.d1/2Sn > Bn Hog log n H —- log log log n) J
Now we formulate the general
THEOREM 18.4 (Orey - Pruitt, 1973). Let a(t) be a nonnegative nondecrtas-
ing continuous function. Then for any d= 1,2,...
tl'*a[t)€WC[\\W[t)\\)
and
if and only if
r feJ\
NON-ACTIVATEDVERSIONwww.avs4you.com
THE LAW OF ITERATED LOGARITHM 195
Remark 1. The function
n
does not satisfy A8.4) if a2 = 2,a3 = d + 2,ak = 2 for 4 < k < n but it does if
an is increased by e > 0 for any n > 2.
It was already mentioned (Chapter 17, Remark 1) that
lim 11^@11 = °° a.s. if d > 3. A8.5)t—>OO
Now we are interested in the rate of convergence in A8.5). This rate is called
rate of escape. We present
THEOREM 18.5 (Dvoretzky - Erdos, 1950). Leta(t) be a nonincreasing, non-
negative function. Then
tl'*a{t)eLLC(\\W(t)\\) (d>3)
and
n^ofrOeLLC^HSnll) {d > 3)
if and only if
f>B"))«-2 < oo. A8.6)n=l
Remark 2. The function
a(x) = (logxlog2 x... (logfcxI+eJdoes not satisfy A8.6) if e < 0, but it does if e > 0.
In the case d = 2 we might ask for the analogue of Theorem of Chung of
Section 5.3, i.e. we are interested in the liminf properties of
m(t) = sup \\W{s)\\ and M{n) = max ||5fc||.
This question seems to be unsolved.
Theorems 18.4 and 18.5 together imply: there are infinitely many n for which
\\Sn\\ > d
and for every n big enough
||S»|| > n^logn)—a±i [d > 3, e > 0). A8.8)
Erdos and Taylor A960/A) proved that if a particle is very far away from the
origin, i.e. A8.7) holds, then it may remain far away forever (d > 3). In fact we
have the following:
NON-ACTIVATEDVERSIONwww.avs4you.com
196 CHAPTER 18
THEOREM 18.6
k>n
NON-ACTIVATEDVERSIONwww.avs4you.com
Chapter 19
Local time
19.1 ?@,n)inZ2The Recurrence Theorem of Chapter 16 clearly implies
n) {- '
hm f@,n) { ..
n-»oosv '
y < oo if a > 3.
Hence we study the limit properties of f@, n) in the case d = 2.
THEOREM 19.1 (Erdos - Taylor, 1960/A). Letd = 2. Then
lim P{f@, n) < x log n} = 1 - e"™
uniformly for 0 < x < (log nK/4 and
()
The proof of this theorem is based on the following:
LEMMA 19.1 (Dvoretzky - Erdos 1950, Erdos - Taylor 1960/A).
>n} = P{?@,n) = 0} = ^ + O((logn)�) (d = 2).
Proof. By A6.4) (cf. also 9. of Notations) we have
n-l
?P{S2fc = 0}P{?@,2n-2A:-2) = 0} = l (n = l,2,...) A9.1)Jk=O
197
NON-ACTIVATEDVERSIONwww.avs4you.com
198 CHAPTER 19
where ?@,0) = 0. Since by Lemma 16.2
P{S2Jk = 0} « (fcTr)�
we have
E !^. A9.2)Jk=O
W
Since the sequence P{?@, 2n) = 0} is nonincreasing by A9.1) and A9.2) we
obtain
1 > P{?@,2n - 2) = 0} ]TP{S2fc = 0} « P{e@,2n - 2) =
*=o"
and
P{e@,2n)=0}<^^. A9.3)
Similarly for any 0 < k < n by A9.1)
/ k \ n~l
1 < EPfe = 0) P(e@,2n - 2k - 2) = 0) + ? PE2j = 0). A9.4)
Thus, if k tends to infinity together with n, A9.4) yields
7T 7T
Taking k = n — [n/ log n] we have
P{?@,2n)=0}>logn
Hence we have the main term of Lemma 19.1. The remainder can be obtained
similarly but with a more tedious calculation.
Proof of Theorem 19.1. Let q = [xlogn] + 1 and p = [n/q\. Then
, n) > xlog n} > P{pq < n} > f[ P Lk -
Pk-X < R |
Assuming that x < logn(log2 n)~l~e by Lemma 19.1 we obtain
P{f@,n) > xlogn} >e-"(l+o(l)) (n-> oo). A9.5)
NON-ACTIVATEDVERSIONwww.avs4you.com
LOCAL TIME 199
In fact we obtain that
P{?@,n) > xlogn} > e~rx{\ + o((logn)-1/4))
uniformly in x for x < (lognK/4.In order to get an upper estimate, let Ek(k = 1,2,... ,q) be the event that
precisely k of the variables p,—
p<_i (i = 1,2,..., q) are greater than or equal to
n, while q— k of them are less than n. Then
n) > xlogn} C
k=l
Hence
O(log nJ
by Lemma 19.1 uniformly in x for x < (lognK/4.A9.5) and A9.6) combined imply the first statement of Theorem 19.1. The
second statement can be obtained similarly observing that by A9.6) and Lemma
19.1 for any x = xra we have
P{f@,n) > xlogn} < exp (-irx + O
Note that Theorem 19.1 easily implies
THEOREM 19.2 Let d = 2. Then
Jim P |pn < exp [-) | = exp(-7rZ)
uniformly for 0 < z < n3/7.
Clearly for any fixed x E Z2 the limit properties of ?(x,n) are the same
as those of f@,n). For example, Theorem 19.1 and Lemma 19.1 remain true
replacing f@,n) by f (x, n). However, if instead of a fixed x a sequence xra (with||xn|| —»• oo) is considered the situation will be completely different. The followingresult gives some information about this case.
NON-ACTIVATEDVERSIONwww.avs4you.com
200 CHAPTER 19
THEOREM 19.3 (Erdos - Taylor, 1960/A). Let d = 2. Then
f 2log||x|| /1 + o/log3|kllogn
if 20 < ||x|| < n1/3,
ilogn
l + O
log.
log
n 1/2
20
Proof. By A6.4) we have
P2»T2n-2»+2 =
i=0
and similarly
,2n) = 0} + ? P{52Jk = x}72n-2Jk+2 = 1
A9.7)
A9.8)Jk=i
provided that x = (xi, x2) with Xi+x2 = 0 (mod 2). A9.7) and A9.8) combined
implyn
,2n) = 0} -Jk=l
Consequently for 1 < kx < k2 < n we get
= 0} - 72n+2 < 72n-2Jk,+2Jk=l
72n-2Jk3+2ik=iki + l
Now in the case 400 < ||x||2 < n2/3 put kxLemmas 19.1, 16.2 and 16.4 we obtain
= \\x\\2 and k2 = [n4/5] then by
P{f(z,2n)=0}-72n+2<>gBn - (log ny)) ? VA:7r
+ °\k*
NON-ACTIVATEDVERSIONwww.avs4you.com
LOCAL TIME 201
i \\
logBn - 2k2)+ O
x
<iogikirlogn
Similarly, for 1 < k3 < n
,2n) .= 0} - l2n+2 > l2n+2 ?>2Jk - P{52JkJk=i
Take
, I'll*3
~~
i ii ii2"log2 ||x||
This, by Lemmas 16.2, 16.4 and 19.1, implies
hence we have Theorem 19.3 in the case 20 < ||x|| < n1/3. The case n1/6 <
n1/2/20 can be treated similarly.As a trivial consequence of Theorem 19.3 we prove
LEMMA 19.2 Let nk = [exp(e*log/E)]. Then for any k big enough
,nf*) - ?@,n4) = 0 | Sr,j = 0,1,2,... ,nk} <3
log A;
Proof. Since \\Snk\\ < nk, for any x e Z2 with ||x|| = nk we have
I>{Z@,nlko*k)-t@,nk)=0\Sj;j = 0,1,2,. ..,nk}
J -
log*"The next theorem gives a complete description of the strong behaviour of
NON-ACTIVATEDVERSIONwww.avs4you.com
202 CHAPTER 19
THEOREM 19.4 (Erdos - Taylor, 1960/A). Let d = 2 and let f{x) resp. g{x)be a decreasing resp. increasing function for which
/(x)logx / oo, g{
Then
n.)) A9.9)
if and only if
f 9\x) r-g(x)j ^ fin irj\J\ x log x
and
f{n) log n e LLC(f@, n)) A9.11)
if and only if
l°°J^-dx < oo. A9.12)J\ xlogx
Remark 1. The function
g{x) = log3 x + 2 log4 x + log5 x + ¦ ¦ ¦ + log^ x + t \o%k+1 x
satisfies A9.10) if and only if r > 1. The function
f(x) = (log log x)�"'
satisfies A9.12) if and only if e > 0.
Proof of A9.9). Instead of A9.9) we prove the somewhat weaker statement
only: for any e > 0
A + ejir-^log n) log3 n G UUC(f@, n)) A9.13)
and
A - ^TT-^log^logan 6 ULC(f@,n)). A9.14)
Let njk = [exp((l + e/Z)k)}. Then by Theorem 19.1
P {^@, nk) > (l + |) ^Oog nk) log3 n
and by Borel - Cantelli lemma
, nk) < fl + |J ^(log nk) log3 njk a.s.
NON-ACTIVATEDVERSIONwww.avs4you.com
LOCAL TIME 203
for all but finitely many k. Let n^ <n < njk+i. Then
f@, n) < f@, Mjk+1) < (l + |^ TT-^log Mjk+1) log3 Mjk+1 < A + ^TT-^log n) log3 M
if k is big enough. Hence we have A9.13).Now we turn to the proof of A9.14). Let
njk =
^Jk = { —r—— > log3 "Jfc
and
Mjk+1
Then clearly Fk C ^7fc+i and by Theorem 19.1 and Lemma 19.2
and similarly for any j < k
Hence we obtain A9.14) by Borel - Cantelli lemma.
Proof of A9.11). Instead of A9.11) we prove the somewhat weaker statement
only: For any e > 0
(lognHloglogn)� e LUC(f@,n)) A9.15)
and
n)-1-' e LLC(f@,n)). A9.16)Let njk = [exp(efc)j. Then by Theorem 19.1
nt)<,. ^(loglognjk
and by Borel - Cantelli lemma
"¦
NON-ACTIVATEDVERSIONwww.avs4you.com
204 CHAPTER 19
for all but finitely many k. Let nk < n < nk+i. Then
jfe logn>
ifcI+e>
(loglogri)i+2e
if k is big enough. Hence we have A9.16).Now we turn to the proof of A9.15). For any r = 1,2,... we have
exp(Cr2r)} = P{p2'-» < exp(Cr2r)}
= P{<;@,exP(Cr2')) > 2'�} » exp (-Since the r.v.'s p2'
—
P2'-1 are independent we obtain
Pv > p-iT~
Pv~x ^ Cr2r i.o. a.s.
and consequentlyf@,exp(Cr2r)) < 2r i.o. a.s.
Let n = exp(Cr2r) with C = log 2. We get
f@, n) < -—: i.o. a.s.
log log n
Hence the proof of Theorem 19.4 (in fact a slightly weaker version of it) is com-
complete.Note that Theorem 19.4 easily implies
THEOREM 19.5 For any s > 0
exp(n(lognI+e)eUUC(pn),MlognI-') GULC(pn),
) GLUC(pn),
exp (A - e)-^— ) e LLC(pn).
19.2 f (n) in Zd
As we have seen (Recurrence Theorem, Chapter 16)
lim f@, n) < oo a.s. if d > 3.
NON-ACTIVATEDVERSIONwww.avs4you.com
LOCAL TIME 205
Similarly for any fixed x G Zd
lim ?(x, n) < oo a.s. if d > 3.n—»oo
v
However, it turns out that
THEOREM 19.6 For any d>l we have
lim f (n) = lim sup ?(x, n) = oo a.s.n—oo n—oo ln—oo
Proof. Theorem 7.1 told us that the length Zn of the longest head run till
n is a.s. larger than or equal to A - e) log n/ log 2 for any e > 0 if n is bigenough. Similarly one can show that the sequence X\, X2, ...,Xn contains a run
«i> -«i, «i,—
«i, ••-,«! of size (l — e) log n/ log 2d. This implies that
minf>n-»oo Jog n 2
which, in turn, implies Theorem 19.6.
A more exact result was obtained by Erdos and Taylor A960/A). They proved
THEOREM 19.7 For any d>3
lim,
= \d a.s.n—oo Jog n
where
\d = -(logP{lim ?@,n) > I})� = -(log(l -7))�.n—*oo
In the case d = 2
— < liminf < limsup — < - a.s.
4tT n—oo (lg)-4 (log n)* 7Tn—oo
Erdos and Taylor A960/A) also formulated the following:
Conjecture. For d = 2
, €(») 1lim T rr = - a.s.
n-^°° (lognj2 7T
NON-ACTIVATEDVERSIONwww.avs4you.com
206 CHAPTER 19
19.3 A few further results
First we give an analogue of Theorem 13.8.
THEOREM 19.8 (Erdos - Taylor, 1960/A).
..1 A _1_ 1
f A ohm 2^ ]= ~ a-5- */ d = 2.
n—oo log nk=x log Pk 7T
The next theorem is an analogue of Theorem 13.1.
THEOREM 19.9 (Erdos - Taylor, 1960/A). Let d = 2,/(n) | oo as (n -> oo)and En be the event that the random walk Sn does not return to the origin between
n and nf^. Then
P(En i.o. ) = 0 or 1
depending on whether the series
oo1
converges or diverges.
Now we turn to the analogue of Theorem 11.27.
THEOREM 19.10 (Erdos - Taylor, 1960/A). Let fr{n) be the number of pointsvisited exactly r times up to n. Then
lim/i(")log'n=1 ^ tf d = 2
and
Jim^-^ =1(l-1)k~1 a.s. if d > 3, k = 1,2,...
where 7 = i(d) is the probability that the path will never return to the origin.
We investigated the favourite values in Section 11.6. We could ask the ana-
analogue questions in the case d > 2. Theorem 19.7 combined with Theorem 19.4
imply that the favourite values are going to infinity in the case d > 2 just like
in the case d = 1 (cf. Theorem 11.26). However, the rate of convergence is not
clear at all. A partial result will be given in Theorem 22.8.
NON-ACTIVATEDVERSIONwww.avs4you.com
Chapter 20
The range
20.1 The range of Sn
Let R{n) be the number of different vectors among Si,S2,...,Sn, i.e. R(n) is
the number of points visited by the particle during the first n steps. The r.v.
•>
will be called the range ofSi,S2,..., Sn. In the case d = 1 Theorem 5.7 essentiallytells us that R{n) is going to infinity like n1/2. In the case d = 2 Theorems 19.1
and 19.4 suggest that R(n) ~ n(logn)�. (Since any fixed point is visited logntimes the number of points visited at all till n is n(logn)�. Clearly it is not a
proof since some points are visited more frequently (cf. Theorem 19.7) and some
less frequently (cf. Theorem 19.10) among the points visited at all.) In fact we
prove
THEOREM 20.1 (Dvoretzky - Erdos, 1950).
EJE(n) =
nn ( n log log n
logn \ (lognJ
lim ?@, n) = 0
limf@,n) =0
limf@,n) = 0
nP
nP
nP
O(logn)
if d = 2,
if d = Z,
if d = 4,
if d>5
B0.1)
207
NON-ACTIVATEDVERSIONwww.avs4you.com
208 CHAPTER 20
where Cd (d = 5,6,...) are positive constants and
/V log log n^ .. ,
** ~n \T~ tf d = 2,I V (log"K /
Vari2(n) = Ei22(n) - (Ei2(n)J < { O(n3/2) ,/ d = 3, B0.2)O(n log n) if d = 4,
O(n) if d>5.
Further, the strong law of large numbers
= 1 a.s. if d > 2. B0.3)
Remark 1. Theorem 5.7 implies that the range does not satisfy the strong law
of large numbers in the case d = 1.
The proof of B0.1) is based on the following:
LEMMA 20.1
P{5n ^ Si fori = l,2,...,n-l}= P{f@,n - 1) = 0}. B0.4)
Remark 2. The left hand side of B0.4) is the probability that the n-th steptakes the path to a new point.
Proof of Lemma 20.1.
= P{Xn + Xn_! + • • • + Xi+1 ? 0 for t = 1,2,..., n - 1}= P{X1 + X2 + --- + Xn-i ^ 0 for t = 1,2,... ,n
- 1}
y ^ 0 for j = 1,2,..., n - 1} = P{?@, n - 1) = 0}.
Hence we have B0.4).Let
(l if Sn^Si for t = l,2,...,n-l,Vn
[0 otherwise.
Then
Jk=i
NON-ACTIVATEDVERSIONwww.avs4you.com
THE RANGE 209
Consequently by Lemma 20.1
EJ2(n) = E ? V* = E PU(°> * - 1) = 0}-Jk=l Jk=l
Hence B0.1) in the case d = 2 follows from Lemma 19.1, and in the case d > 3
it follows from Lemma 16.7.
In order to prove B0.2) we present two lemmas.
LEMMA 20.2 Let 1 < m < n. Then
Proof.
^ 5m, i = 1,..., m - 1; Sj ? Sn, j = 1,..., n -
5m, t = 1,..., m - l}P{5y ^ Sn, j = m,..., n - 1}
Sm, i = 1,..., m - l}P{Sy ^ 5n_m+i, j = 1,... n - m}
LEMMA 20.3
Vari2(n) < 2Ei2(n) (e72 (n - [^]) - EJ2(n)
Proof. By Lemma 20.2
Vari2(n) =
<2 E
2
Since by Lemma 20.1 E^y is nonincreasing the max is attained for t = [n/2] + 1
and Lemma 20.3 is proved.Then B0.2) follows from Lemma 20.3 and B0.1). B0.3) can be obtained by
routine methods.
Donsker and Varadhan A979) were interested in another property of R(n).In fact they investigated the limit behaviour of Eexp(—vR(n))(v > 0, n —> oo).They proved
NON-ACTIVATEDVERSIONwww.avs4you.com
210 CHAPTER 20
THEOREM 20.2 For any u > 0 and d = 2,3,...
lim n~*+* logE(exp(-i/i2(n)) = -k{u)
where
and at is the lowest eigenvalue of —1/2A for the sphere of unit volume in Rd
with zero boundary values.
Remark 3. In the case d = 2 Theorem 20.1 claims that R{n) is typicallyTrn/logn. Hence we could expect that Eexp(—uR(n)) ~ exp(—i/irn/logn).However, Theorem 20.2 claims that Eexp(—uR{n)) ~ exp(—k{u)nll2). Compar-Comparing these two results it turns out that in the asymptotic behaviour of Ee~"^n^the very small values of R(n) contribute most. This fact is explained by the
following:
LEMMA 20.4 For any v > 0 there exists a Cv > 0 such that
E(exp(-i/JE(n))) > exp(-Cun1/2).
Proof.
E(exp(-i/JE(n))) > E(exp(-i/iE(n)) | Mn < nl'A)Y{Mn < n1/4)> exp(-i/?rn1/2)P(Afn < n1/4).
By Theorem 17.5
P(Mn<n1/4)=exp(-O(n1/2))and we have Lemma 20.4.
20.2 Wiener sausage
Let W(t) € Rd be a Wiener process. Consider the random set
Br{T)=U0<t<T(W{t)+Kr)= {x: xe Rd,x = W(t) + a for some 0 < t < T and a € Kr}
where
Kr = {x: \\x\\ < r}.
Br(T) is called Wiener sausage. The most important results are summarized in
NON-ACTIVATEDVERSIONwww.avs4you.com
THE RANGE 211
THEOREM 20.3 (cf. Le Gall, 1988).
lim !^A(Br(T)) = 2tt a.s. B0.5)
for any r > 0 and d = 2. If d > 3 then
WmT-lX{Br{T))=c{r,d) a.s. B0.6)T—>oo
where c(r,d) is a positive valued known function of r and d. Further,
lim P{Kd(T)(X(Br(T)) - Ld{T)) < x} = $(ai + 0) B0.7)T—*oo
where
,2tt- if d = 2,
'
l{d,r)T if d>3,
{^P- H d = 2,
(TlogT)�/2 if d = Z,T*~*/2 if d > 4
and a,/?,7 are known functions of r and d.
Clearly B0.5) and B0.6) are strongly related to B0.1) and B0.3). In fact
one can define the random walk sausage as
Jk=O
Then Theorem 20.1 implies
lim l^-\{Bls){N))=ir2r2 a.s. if d = 2N-00 N
and
lim N^XiB^iN)) = r27rP{ lim f@,n) = 0} a.s. if d > 3.
The analogue of B0.7) is unknown for random walk sausage but we can
formulate the following:
NON-ACTIVATEDVERSIONwww.avs4you.com
212 CHAPTER 20
Conjecture 1.
lim P {K{dS)(N)(X(Bls\N)) - L{fJ(N)) < x} = *N—KX3 (
where
(^^L if d = 2,log AT
r27rP{ lim f@, n) = 0}N if d > 3
and K, a, C are suitable normalizing constants.
NON-ACTIVATEDVERSIONwww.avs4you.com
Chapter 21
Selfcrossing
It is easy to see (and Theorem 20.1 also implies) that the path of a random walk
crosses itself infinitely many times for any d > 1 with probability 1. We mean
that there exists an infinite sequence {Un,Vn} of positive integer valued r.v.'s
such that S(Un) = S(Un + Vn),a.nd0<U1<U2<...,{n = l,2,...). However,we ask the following question: will selfcrossings occur after a long time? For
example, we ask whether the crossing S(Un) = S(Un + Vn) will occur for everyn = 1,2,... if we assume that Vn converges to infinity with a great speed and Unconverges to infinity much slower. In fact Erdos and Taylor A960/B) proposedthe following two problems.
Problem A. Let f(n) | oo be a positive integer valued function. What are the
conditions on the rate of increase of f(n) which are necessary and sufficient to
ensure that the paths {So, Si,..., Sn} and {?„+/(„), ?„+/(„)+i;...} have points in
common for infinitely many values of n with probability 1?
Problem B. A point Sn of a path is said to be "good" if there are no pointscommon to {S0,Si,...,Sn} and {5rn+1,5rn+2,...}. For d = 1 or 2 there are no
good points with probability 1. For d > 3 there might be some good points: how
many are there?
As far as Problem A is concerned we have
THEOREM 21.1 (Erdos - Taylor, 1960/B). Let f(n) | oo be a positive integervalued function and let En be the event that paths
{S0,Si,...,Sn} and {Srn+/(n)+1,5rn+/(n)+2,...}have points in common. Then
(i) for d = 3, if f(n) = n(<p(n)J and <p(n) is increasing, then
P(En i.o.) = 0 or 1 B1.1)
213
NON-ACTIVATEDVERSIONwww.avs4you.com
214 CHAPTER 21
depending on whether Efc^i((PBA:)) lconverges or diverges,
(ii) for d — 4, if f(n) = nx{n) and x{n) I5 increasing, then we have B1.1)depending on whether E^=i(^xBA:))� converges or diverges,
(iii) for d> 5, if
supm>n rn n
(for some C > 0) then we have B1.1) depending on whether
n=l
converges or diverges.
An answer of Problem B is
THEOREM 21.2 (Erdos - Taylor, 1960/B). For d>Z let G^(n) be the num-
number of integers r (l < r < n) for which (Sq, Si, ..., Sr) and (Sr+i, Sr+2, ¦ ¦ •) have
no points in common. Then
(i) d = 3. For any e > 0
P{GC)(n) > n1/2+e i.o. } = 0.
(ii) d = 4.
P{0 = liminf" rr""'< limsup- ^)log"< C) = 1.n—»oo fi n—>oo M J
(iii) d > 5.
hm i—- = t& a.s.n—oo n
where Tj is an increasing sequence of positive numbers with r^ f 1 as d —> oo.
Remark 1. Applying Theorem 21.1 for d = 4 and /(n) = n — 1 we find that for
infinitely many n the paths {So, ^i, • • • > Sn} and {^n, 52n+i, • • •} have a point in
common. This statement is not true for d > 5.
Our next theorem is intuitively clear by Remark 1.
THEOREM 21.3 (Erdos - Taylor, 1960/B, Lawler, 1980). For d = 4, two inde-
independent random walks which start from any two given fixed points have infinitelymany common points with probability 1; whereas for d > 5, two independentrandom walks meet only finitely often, with probability 1.
NON-ACTIVATEDVERSIONwww.avs4you.com
SELFCROSSING 215
Remark 2. Theorem 21.3 tells us that the paths of two independent random
walks in Zd (d < 4) cross each other. It does not mean that the particles meet
each other.
One can also investigate the selfcrossing of a <i-dimensional Wiener process.
Dvoretzky - Erdos - Kakutani A950) proved the following beautiful theorem:
THEOREM 21.4 For d < 3 almost all paths of a Wiener process have double
points (in fact they have infinitely many double points), i.e. there exist r.v.'s
0 < U < V < oo with W(U) = W{V). For d > 4 almost all paths of a Wiener
process have no double points.
Remark 3. Comparing Theorems 21.1 and 21.4 in the case d = 4 we obtain that
for infinitely many n the paths {So, • • •> $*.} and {^n, 52n+i, • • •} have a point in
common, while for any t > 0 the paths {W(s);0 < s < t} and {W(s);2t <
s < oo} have no points in common with probability 1. This surprising fact is
explained by Erdos and Taylor A961) as follows: for d = 4 with probability 1
the paths {W^(s);0 < s < t) and (W^s); 2t < s < oo} approach arbitrarily close
to each other for arbitrarily large values of t. Thus they have infinitely many
near misses, but fail to intersect. This explanation suggests the following:
Question 1. Characterize the set of those functions /(•) for which
)||=O a.s. (d = 4).2l<u
NON-ACTIVATEDVERSIONwww.avs4you.com
Chapter 22
Large covered balls
22.1 Completely covered discs in Z2
We say that the disc
Q{r) = {x e Z\\\x\\ < r)
is covered by the random walk {Sk} in time n if
?(x, n) > 0 for every x G Q{r).
Let R{n) be the largest integer for which Q(R(n)) is covered in time n. The
Recurrence Theorem of Chapter 16 implies that
lim R(n) = oo a.s. B2.1)
We are interested in the rate of convergence in B2.1). We prove
THEOREM 22.1 (Erdos - Revesz, 1988, Revesz, 1989/A-B, Auer, 1990). For
any s > 0 and C > 0 we have
exp B(log nI/2 log3 n) 6 UUC(i2(n)), B2.2)
exp (^^(lognloganI/2) 6 ULC(R(n)), B2.3)
exp (C(lognI/2) e LUC(i2(n)), B2.4)
exp ((log nI/2(log log n)�/2"') € LLC(i2(n)). B2.5)
About the limit distribution of R(n) we prove
217
NON-ACTIVATEDVERSIONwww.avs4you.com
218 CHAPTER 22
THEOREM 22.2 (Revesz, 1989/A, 1989/B). For any z > 0
< limsupPv » v "
> z \ < exp — . B2.6)~
n-oo I logM J\ 4 I
This theorem suggests the following:
Conjecture 1. There exists a A > 0 such that
>,}«p(logn J
Note that we cannot prove even the existence of the limit distribution.
At first we introduce a few notations and prove some lemmas.
Let a(r) be the probability that the random walk {Sn} hits the circle of radius
r before returning to the point 0 = @,0), i.e.
a(r) = P{inf{n : ||5n|| > r} < inf{n : n > l,Sn = 0}}.
Further, let
p@ ~> x) = P{inf{n : n > l,Sn = 0} > inf{n : n > l,Sn = x}}= P{{5rn} reaches x before returning to 0}.
LEMMA 22.1
J|im a(r) log r = tt/2. B2.7)
Proof. Clearly we have
{inf{n : ||5n|| > r} > inf{n : n > l,Sn = 0}}
C {€@, r2 log2 r) > 0} + ( max ||5*|| < r) .
^0<Jk<r3log''r J
Since by Lemma 19.1
P{f@,r2log2r) =0}«7r/2logr
and by a trivial calculation (cf. Theorem 17.5)
P{ max ||Sfc||<r}=-o(l/logr),0<Jfc<r3log3r
NON-ACTIVATEDVERSIONwww.avs4you.com
LARGE COVERED BALLS 219
we have
«(') >2 log r
Observe also
a(r) <P{ max ||54|| > r} + PU^logr)�) = 0}.0<Jk<r3(logr)-1
Since by Theorem 17.5
P{ max ||5*|| >r} = exp(-O(logr)),CKJKr^logr)-1
applying again Lemma 19.1 we obtain B2.7).
Remark 1. Lemma 22.1 is closely related to Lemma 17.1.
LEMMA 22.2 There exists a positive constant C such that
Q
log||x||
for any x € Z2 with \\x\\ > 2. Further,
— < liminf p@ ^ x) log ||x|| < limsupp@ /v> x) log ||x|| < -.
12 ||x|| — oo ||x||—oo 2
Proof. Let x = ||x||e'*\ Then by Lemma 22.1 the probability that the particlecrosses the arc ||x||e'* (<p — tt/3 < if) < (p + tt/3) before returning to 0 is largerthan (l - eOrFlog Hxll)� (for any e > 0 if ||x|| is big enough). Since startingfrom any point of the arc ||x||e'* (<?> — tt/3 < xjj < (p + tt/3) the probability that
the particle hits x before 0 is larger than 1/2 we obtain the lower estimate of the
liminf. The upper estimate is a trivial consequence of Lemma 22.1.
Spitzer A964) obtained the exact order of p@ /v> x). He proved
LEMMA 22.3 (Spitzer, 1964, pp. 117, 124 and 125).
7T + OA) ..
p@ ~x) =
4lo7Has l|x|l^°°-
LEMMA 22.4 Let
Yi = t{x\pi)-t{x,pi-l)-l (t = l,2,...)
NON-ACTIVATEDVERSIONwww.avs4you.com
220 CHAPTER 22
and Zi = —Yi. Then there exists a positive constant C* .such that for any 6 > 0
and f(n) f oo for which n/ f(n) —> oo we have
(c21/2 / \ !/2\
—^~ + c vTRJ B2*8)
l/2>
i + Z2 + ¦ ¦ ¦ + Zn > Son3'4} < exp [ -^- + C*[Tf^) } B2.9)
where
o* = -EY* = 2p{Q^x)
^
( n^ \Ikll < exP ( 77TT) • B2-10)
B2.8) and B2.9) combined imply
. B2.11)
Proof. By a simple combinatorial argument (cf. Theorem 9.7) we get
= k} = (l-p@-x))fc(p@- x)J (A: = 0,1,2,...),= 0,
and
/xm = EV- = (-1)", + f; kmp2qkJk=O
2qk< (-l)mq + Ep2qkk{k + l)--'{k + m-l) = {-l)mqJk=i
where p = p@ /\> x) and ^ = 1 —
p. Hence
|pm| <q + m\pl-mq (m
NON-ACTIVATEDVERSIONwww.avs4you.com
LARGE COVERED BALLS 221
Similarly
Kl = \EZ?\ < q + mlpl-mq (m = 3,4,.. .)•
Let
2
g(z) = EexpBy1) = qe~z +P
+z,
tp(z) = \og g{z)
and
s = sn= 6o-ln-ll\
Then by Lemma 22.2 and condition B2.10) we have
2
if n is big enough. Hence
(k = 3,4,...) and
\*{s)\ < log2 2 oo Je oo
f fJk=3
K-Jk=3
3 . MM ^s3V+M - 2 <—- + 3< log I 1 H h qs e + pq I - I 2
2' *~
" ' "
Vp7 "/"
2" " "
P2
Similarly, r* M I ,/ M
<^2S2 3 S2S3
llogEexp^ZJl = \v{-s)\ < —— + s e + —-.
L p
Let
Fn(k)=-P{Y1 + Y2 +¦¦¦ + ?„ = k}and define a sequence l/j, U2, • • • of i.i.d.r.v.'s with
pm _ ^.\ — e~^a)gak'p{Yi = k\.
Then
NON-ACTIVATEDVERSIONwww.avs4you.com
222 CHAPTER 22
where
Hence
Yn > Son3"}
< exp(nip(s) - s6an3/A)2s3'
< exp n
and we have B2.8).Note that the proof of B2.9) is going on the same line but instead of the
sequence {Un} we have to use the sequence {U^} defined by
; = k} = e-*(-V*P(Z! = k)
and we have the lemma.
Now we turn to the proof of B2.5). In fact we prove the much stronger
THEOREM 22.3 (Auer, 1990). For any e > 0 we have
lim sup li,, -II =0 a.s. B2.12)—
|,||<r.'^0 n)
where
rn = exp
Remark 2. Note that B2.5) claims that the disc around the origin of radius
rn is covered in time n. The meaning of B2.12) is that the very same disc
is "homogeneously" covered, i.e. every point of this disc will be visited about
f@, n) ~ log n times during the first n steps. For the one-dimensional analogueof this theorem, cf. Theorem 11.9.
Proof of Theorem 22.3. Clearly
i{x,Pn)i =
y1 + y2 + --- + yw
NON-ACTIVATEDVERSIONwww.avs4you.com
LARGE COVERED BALLS 223
and by Lemma 22.4 we have
P { sup
•
supll*ll<An
sup n\ ><5<rn3/4}
n*
exp-
2 +C\f(n),where An = exp(n1/2//(n))- Let f(n)lemma we obtain
= (lo8n)e then by the Borel " Cantelli
sup -l|=0 a.S.
which, in turn, implies
lim sup supn~>0°
||*||<An Pn<m<Pn+l
= 0 a.s.
Then replacing n by ^@, N) and recalling that
for all but finitely many N (cf. Theorem 19.4) we obtain the theorem.
In order to prove B2.2) of Theorem 22.1 we prove two lemmas and introduce
a few notations. Let
J'nJ-\0 if Z(x,n)=0,mk = mk{xu x2,.. •, xk; n) = E{I{xu n)I{x2, n)..., I{xk, n))
= P{e(x!,n) >0,e(x2,n) >0,...,e(xfc,n) > 0},
^B) = ux = min{A;, k > 0, Sk = 2},
t=i
Then we have
NON-ACTIVATEDVERSIONwww.avs4you.com
224 CHAPTER 22
LEMMA 22.5 For any 0 < q < 1 k = 2,3,... we have
mi(u;n-qn)[M{qn) + A - k)mk{xu x2,... ,xk; qn)]. mi(v,n)M(n)
":"'(t)()
where u € Z2 and v 6 Z2 are defined by
mi(v,n) = max mi(x,,n) and mi(u,n) = min mi(x,,n).
In order to present the proof in an intelligible form we prove Lemma 22.5 first
in the case k = 2. That is, we prove
LEMMA 22.6 For any 0 < q < 1 we have
xiqn) +m1(y;qn) - m2{x,y;qn)\
n) <.
l + mi(x-y;n)
Proof.
m2{x,y;n) = P{/(x,n) = 1, J(y,n) = 1}
Cx.n) = l,/(y,n) = 1 | ux = k < uv}Y{ux = k < uv}Jk=O
n
J^ P{/(x, n) = 1, J(y, n) = 1 | uy = k < ux}V{uy = k < ux}
.n) = 1 I ux = k < uy}Y{ux = k<uv}
k=0
Jk=O
Jk=O
n
= ]P P{^(y —
x,n— k) =
k=o
n
+ J^I>{I{x-y,n-k) =
k=o
Consequently we have
(x -y,n) = 1}P |f:(K = *<i/,) + (i/,=*< i/,))}U=o J
= P{/(x -
y, n) = 1}P{/(*, n) = 1 or I(y, n) = 1}= m^x- y;n)[mi{x;n) +m1(y,n) - m2(x, y,n)}
NON-ACTIVATEDVERSIONwww.avs4you.com
LARGE COVERED BALLS 225
which implies the upper part of Lemma 22.6.
We also have
™2{x,y;n) > ?P{/(y -
x,n- k) = l}P{i/x = * < uy}
qn
T.Jk=O
+ 5Z p{^(x ~
j/' n ~ ^) =
Jk=O
> P{/(x -
y, n - gn) = l}P{/(x, qn) = 1 or J(y, gn) = 1}= m^x- y;n-qn)[mx{x;qn) +mi{y;qn) - m2(x,y;qn)}.
Hence we have Lemma 22.6.
Proof of Lemma 22.5. Let Pk resp. Pjk(r) be the set of permutations of the inte-
integers 1,2,..., k resp. 1,2,... ,r-l,r+l,..., k. Further, let A = A(ii,i2,...,ik;j) =
{u{Xil) < i/{Xii) < ..., u{xik_l) =j< v{xik)}- Then we have
mk{xi,x2,...,xk;n) = ^ P{/(x!,n) =...
= I(xk,n) =
Consequently
= m!(v,n)[.M(n) + A
which implies the upper part of the inequality of Lemma 22.5.
We also have
(U •¦
<
NON-ACTIVATEDVERSIONwww.avs4you.com
226 CHAPTER 22
r=l
= P{/(u,n-gn) =
xP
= m^u.n- gn)[.M(gn) + (l - k)mk\.Hence we have Lemma 22.5.
Let N = N(n) / oo and k = k(n) / oo be sequences of positive integerswith N(n) < n1/3. Assume that there exists an e > 0 for which k(n) < (N(n))e.Then for any n = 1,2,... there exists a sequence X\ = Zi(n), x2 = x2(n),..., x^ =
xk{n) € Z2{k = k(n)) such that
N -1 < \\xi\\ <N (i = l,2,...,Jk),Nl~e < \\x{ - Xj\\ < N A < i < j < k).
Now we formulate our
LEMMA 22.7 For the above defined X\, x2,... xk we have
(^^( (^^jj). B2.13)
Further, if
N{n) = exp((log nI/2 log3 n) and k{n) = exp((log nI/2) B2.14)then
i, x2,..., xk; n) < exp(-B - 4e) log3 n). B2.15)
Proof. By Lemma 22.5 and Theorem 19.3 we have
logn
NON-ACTIVATEDVERSIONwww.avs4you.com
LARGE COVERED BALLS 227
Hence we have B2.13). If B2.14) holds then
| 0(logn I I logn
< exp I-B - 4g)lo n
flog") ' 1 = «P("B - 4e) log3
and we obtain B2.15).Now we can present the
Proof of B2.2). Let N{n) be defined by B2.14) and let
ni = (exp(eJ)],
N{n) =expB(lognI/Mog8n),^•+1=exp((logni+1I/2).
Then
where x,- = x,(ni+1) (i = 1,2,..., kj+l). Hence by the Borel - Cantelli lemma
a.s.
for all but finitely many j. Let n, < n < nJ+1. Then
R(n) < R{nj+1) < N{n3) < N(n)
which proves B2.2).Since B2.4) is a trivial consequence of the upper inequality of B2.6) we prove
the upper part of B2.6). In fact we prove a bit more:
THEOREM 22.4 (Revesz, 1989/A). For any z > 0 and for any n = 2,3,...we have
y B2.16)4/
In order to prove Theorem 22.4 let
M = M{n) = exp(C(log n)l'2) (C > 0)
NON-ACTIVATEDVERSIONwww.avs4you.com
228 CHAPTER 22
and
K = K{n) = <T exp (^-(lognI/2) (C* > 0).
Then for any n = 1,2,... there exists a sequence yx = j/i(n), y2 = 5/2A),..., yK
I/if(n) such that
M-\< \\vi\\ <M (i = l,2,...,K),
MZ'A < \\Vi - yy|| < M A < 1 < j < K).
if C* is small enough. Now we formulate our
LEMMA 22.8
mK(yi,y2,...,yK;n) < exp I——I .
Proof. In the same way as we proved Lemma 22.7 we obtain
«-(-?Hence we have Lemma 22.8.
Proof of Theorem 22.4. Clearly
P{R{n) > exp^QognI/2)} < mK < exp I-—1
which proves Theorem 22.4.
Instead of proving B2.3) we prove the following stronger
THEOREM 22.5 For any 0 < 0 < (tt/ 120I/2,9/10 < <52 < 1 and e > 0 we
have
inf ?(x, n) > -—pr^-(lognlog3 nI/2 t.o. a.s.
I<M)V ~
V^
where
@A — e)—7=v71"
NON-ACTIVATEDVERSIONwww.avs4you.com
LARGE COVERED BALLS 229
In order to prove Theorem 22.5 let pi@ ^ x),p2@ ^ x),... resp. pi(x /v>
0),p2(x ^* 0),... be the first, second, ... waiting times to reach x from 0, resp.
to reach 0 from x, i.e.
Pi@ '\> x) = inf{n : n > 1, Sn = x},
Pi(x '\> 0) = inf{n : n > Pi@ ^ x), Sn = 0} - pi@ "^ x),
p2@ '\> x) = inf{n : n > />i@ ^ x) + ^(x ^ 0), 5n = x}
p2(x '\> 0) = inf{n : n > p^O '\> x) + Pi(x '\> 0) + p2@ '\> x), 5n = 0}- (Pi@ ~> x) + pi(x ~f 0) + p2{0 ~f x)),...
Let r@ /v> x, n) be the number of 0 /v> x excursions completed before n, i.e.
f 1r@ '\> x, n) = max <» : ^!(Pj@ /v> x) + py(x 'v> 0)) + p,-@ 'v> x) < n > .
I ;=i J
In the proof of Theorem 22.5 the following lemma will be used.
LEMMA 22.9 For any 0 < 0 < (tt/120I/2,9/10 < 62 < 1 and n big enoughwe have
Pin�/2 inf r@^x,,n)<l-<5}<exp(-^^V B2.17)||i||<e«vs \ 60 0 /
(For pn, see Notation 6.)
Proof. Let q=l — p=l— p@ /v> x). Then applying Bernstein inequality(Theorem 2.3) with e = 6p and Lemma 22.2 we obtain
p |r((Wn
n
60log||x|
provided that ||x|| is big enough.Hence
inf n-^2T@^x,pn)<l-6}<-pl infr(° ^ ^^)
^ 1 _
Anp
60 0B2.18)
NON-ACTIVATEDVERSIONwww.avs4you.com
230 CHAPTER 22
which implies B2.17).
Proof of Theorem 22.5. B2.17) clearly implies that
limmfn�/2 inf r@ ~> x,pn) > 1 - 6
for any 0 < 0 < (tt/120I/2 and 9/10 < 62 < 1.
Observe that Theorem 19.5 and B2.18) imply
inf rfcWs.expf*1/^™)) >1-S i.o. a.s.,
i.e.
inf r@ 'v> x,n) > —p^(lognlogonI/2 i.o. a.s.
||||<M)V ~
^T
which in turn implies Theorem 22.5.
Now, we have to prove the lower inequality of B2.6). In fact instead of provingthe lower part of B2.6) we prove the following stronger
THEOREM 22.6 For any e > 0 and z > 0 there exists a positive integerNo = N0(e,z) such that
P{ inf r@^x,n)>(l-<5)BlognI/2}>exp(-7T2)-e B2.19)||i||<exp(e(zlogn)l/2)
if n > N0,0 < 0 < (tt/120I/2 and 9/10 < <52 < 1.
Proof. Theorem 19.2 and B2.18) imply that for any e > 0 and z > 0 there
exists a positive integer No = No(e,z) such that
P{n~1/2 inf r@ ^ x, pn) < 1 - <$} < e
||i||<e«v^
and
P \pn < exp l-J I > exp(-7rz) - e
if n > iV0. Consequently
p( inf rfo-^x^xpf-)) ^(l-^n1/2}
( inf rfo^x^xpf-)) > (l-<5)n1/2,pn<exp(-)le^ \ \Z) ) Z JZ J
> Pi inf r{0^x,pn) >(l-<5)n1/2,pn<exp(-)|> P{ inf T@^x,pn)>(l-6)nl'2}-I>{pn>exp(-)}
||x||<e®v^ I \zJ J
> 1 — e — A — exp(—7T2)) — e = exp(-Kz) — 2e
NON-ACTIVATEDVERSIONwww.avs4you.com
LARGE COVERED BALLS 231
if n > No.Hence we have Theorem 22.6 as well as Theorems 22.1 and 22.2.
Theorem 22.3 clearly implies that for any fixed x € Z2
lim B2-20)
It is worthwhile to note that for fixed x in B2.20) a rate of convergence can
also be obtained. In fact we have
THEOREM 22.7
where a(x) is a positive constant depending only on x.
Remark 3. Theorem 22.7 and Theorem 19.4 combined imply
lim( log n
n-°° \(loglognI+ey ?@, n)= 0 a.s.
Before presenting the proof of Theorem 22.7 introduce the following notations:
let Hi, H2,... be the local time of 0 during the first, second, ... 0 '\> x excursions,i.e.
= 52 = x) + pi{x ~e 0) Si,
where
i?3 = ^@ ^ x) + pi{x ~c 0) + p2@ ~c x) + p2(x ~c 0) + p3{0 ~c x).
Observe that H^ H2,... are i.i.d.r.v.'s with distribution
P(S1 = *)=P(?@,Pi((W *))=*)
= A - p@ -^ ijj^-^O -^ x) (* = 1,2,...).
Consequently
S! - (p@ - x))�J = (p@ - p@ - x))
NON-ACTIVATEDVERSIONwww.avs4you.com
232 CHAPTER 22
and
5,'n^ BnloglognI/2
-
p((W z)
where
z) + pi(z ^ 0)),x)+ Pi{x ^ 0) + p2@ ~> x) + p2(z ~> 0)) -
Since
S1 + s2 + ... + ST((K.x,n) < ?@, n) < Ex + 52 + ¦ ¦ ¦ + ST@^
and the sequence r@ /v> x, n) takes every positive integer we have
imSUP7;r;. . , ,,= 2 ;r
n-oo Br@ ^ z,n) log log r@ ^ XjU)I/2 p@ ^ x)
By the law of large numbers
a.S.
n—oo
Hence
1T-.S<SPB?@,n)loglog{@,n))'A=( p@ -«) j ='«*' «*
and we have Theorem 22.7.
Remark 4. Theorem 11.26 claimed that the favourite values of a random
walk in Z1 converge to infinity. It is natural to ask the analogue question in
higher dimension. In the case d > 3 the Polya Reccurence Theorem implies that
the favourite values are also going to infinity. Comparing Theorem 19.4 and
Theorem 19.7 we find that in Z2 the favourite values are also going to infinity.Theorem 22.3 also says that the rate of convergence is not very slow. In fact we
get
THEOREM 22.8 Let d = 2 and consider a sequence {xn} for which ?(xn, n) =
f (n). Then for any e > 0 we have
liminr rj- >1/,/1—: N .,, .= oo a.s.
"-<» exp((lognI/2(loglogn)-1/2-«)
NON-ACTIVATEDVERSIONwww.avs4you.com
LARGE COVERED BALLS 233
Remark 5. Replacing the random walk in Theorems 22.1 and 22.2 by a Wiener
sausage Br(T) one can ask whether these theorems remain valid. A harder
question is to study the case where r = rT | 0. In fact Theorem 17.4 impliesthat if rT < T~^ogT>>' (with some e > 0) then the analogues of Theorems 22.1
and 22.2 cannot be true anymore.
22.2 Discs covered with positive density
The results of Section 22.1 claimed that the radius of the largest covered disc
is about exp((lognI/2). Now we are interested in the relative frequency of the
visited points in a larger disc.
In order to formulate our results, introduce the following notations:
I(xn)-il if ^'n)>0''^-[O if e(*,n)=0,
K(N,n) = (N2*)-1 ? I(x,n);zeq(N)
i.e. K(N,n) is the density (relative frequency) of the points of Q{N) covered bythe random walk {Sk,0 < k < n).
Our first theorem claims that if we consider the disc of radius exp((logn)a)(a < 1) or even of radius exp(logn(loglogn)~2~e) (e > 0) then the density of the
covered points converges to one a.s. In fact we have
THEOREM 22.9 (Auer - Revesz, 1989). For any e > 0
=l a.s.
Proof. Consider
where
N = Nn = exp
Then by Lemma 22.2 we have
NON-ACTIVATEDVERSIONwww.avs4you.com
234 CHAPTER 22
provided that ||x|| < N. Hence
± E(l-/(x,pn))<exp(-C(lognI+e),
and by the Markov inequality for any 6 > 0
P{1 - K(N,pn) >S}< S-'
which, in turn, by Borel - Cantelli lemma implies that
Jim A - K{N,pn)) = 0 a.s. B2.21)n—»oo
Let m = mn = [exp(n(lognI+e)]. Then by Theorem 19.5 mn > pn a.s. for
all but finitely many n. Hence B2.21) implies
\im(l - K(N,m)) =0 a.s.
Observe that given the choice of m and N we have
\ogm \N
( log m
J 6XP
and we obtain
f-—(
limKlexpf-— TiT7)'m)=1 a-s-n^°°
\ \(loglogmJ+«/ /
Consequently we also have
1- TS I ( l°g"Wl \ \ ,
hm K exp -—: rz-r- , mn =1 a.s.»»« v vA°g1°gm) / /
This proves the theorem.
Theorem 22.9 tells us that almost all points of the disc Q(exp(log n)a)(l/2 <
a < 1) will be visited by the random walk {So, Si,..., Sn}. At the same time byTheorem 22.1 we know that some points of Q(exp((logn)a)) will surely not be
visited. We can ask how many points of Q(exp((logn)a)) will not be visited, i.e.
what is the rate of convergence in Theorem 22.9? However, it is more interestingto investigate the geometrical properties of the non-visited points. For example:what is the area of the largest non-visited disc within Q(exp((logn)a))? Bynon-visited disc we mean a disc having only non-visited points. The followingtheorem claims that with probability 1 there exists a non-visited disc of radius
NON-ACTIVATEDVERSIONwww.avs4you.com
LARGE COVERED BALLS 235
exp((logn)^) within the disc Q(exp((logn)a)) for every /3 < a provided that
a > 1/2.Let
Q(u,r)={x€Z2,\\x-u\\<r}.
Then we have
THEOREM 22.10 (Auer - Revesz, 1989). Let
1/2 < a < 1 and 0 < a.
Then there exists a sequence of random vectors Ui, u-i,... such that
Q(un,exp((logn)^)) C Q@,exp((logn)a)) = Q(exp((logn)a))
and
I{x,n)=0 for all x € Q(un,exp((logn)^)).
In order to prove Theorem 22.10, first we introduce a notation and present two
lemmas.
Let N > 0 and ux, u2,..., uk € Z2(k = 1,2,...) be such points for which the
discs
A = 1,2,...,*)
are disjoint. Denote by
mk{QuQ2,...,Qk-,n) = P{Vi = 1,2,...,/:, 3y, € Q, such that /(y,,n) = 1}
the probability that the discs Q\,Qi,---,Qk are visited during the first n steps.
LEMMA 22.10 Let
exp((logn)a) < ||rx|| < n1/3 N <e^
and
O<0 < a< 1.
Then
for a suitable constant C > 0.
NON-ACTIVATEDVERSIONwww.avs4you.com
236 CHAPTER 22
Proof. It is easy to see that
P{/(u,2n) = l}>m1(Q(u,iV);n) min P{?(u,2n) - f(u,n) > 0 | Sn = y}.y€Q{uN)
Hence by Theorem 19.3
mi(Q(u,N);n) < )r^-{ < 1 - C(logn)-'.
fe)Hence Lemma 22.10 is proved.
LEMMA 22.11 Let
(i) 0 < 0 < a < 1,
(ii) iV=exp((lognn,(iii) ui, u2,..., ujt € Z2 be a sequence for which
\\ui\\<n1"- A = 1,2,...,*),
||u, - tty|| > exp((logn)a+e) A < i < j < k).
Then
rnk{QuQ2,...,Qk;n) <zxv{-C{logn)a-l
for a suitable C > 0 where Q, = Q(ui,N) (i = 1,2,..., k).
Proof. Clearly
= P-fujL^VQy are visited before n and Qt- is the last visited disc}}< P{U*=1{the discs Qi,..., Q,-i, Qi+i, ¦ -iQk are visited before n}}x maxmaxPlQ. is visited before 2n\ Sn = x).
ijtj xeQ;
Hence by Lemma 22.10
mk{Qi,...,Qk;n)
< (ibmi<-i{Qi> ¦ ¦ -'Qi-uQi+u ¦ ¦ -,Qk;n) - {k - l)mfc(Ql5... ,Qk;n)
x(l-C(logn)a-1)
NON-ACTIVATEDVERSIONwww.avs4you.com
LARGE COVERED BALLS 237
and
mk{Qi,...,Qk;n)¦¦¦,Qk]n)
Since1-a
for any 0 < a < 1 and Jk>lwe have
mk{Q1,...,Qk;n)
by induction
mk{Q1,...,Qk;n) <
exp (-dlogn)*-1^) <
V t=2 */
and we have Lemma 22.11.
Remark 1. Lemma 22.11 is a natural analogue of Lemmas 22.5 and 22.7.
Proof of Theorem 22.10. Let
1/2 < a< a + e < 1, 0 < a
and
iV = exp((logn)/?).Then there exist k = k(n) = exp((log n)a) points Ui, u^,..., uk such that
||u,||<exp((logn)a+e) (i= 1,2,...,*),
and
\\xi - xy|| > exp((logn)a+e/2) (i,i = 1,2,..., k; i / j).
Then by Lemma 22.11
mk{QuQ2,...,Qk;n) < exp(-C(lognJ"�)
NON-ACTIVATEDVERSIONwww.avs4you.com
238 CHAPTER 22
where Qi = Q(ui,N). Choosing n = n, = e3 the Borel - Cantelli lemma impliesthat with probability 1 at least one Q, (i = 1,2,..., k(rtj)) is not visited till nifor all but finitely many j.
Let rij < n < rij+i. Then with probability 1 there exists a u with ||uj| <
exp((> + l)a+e) such that the disc Q{u,N) {N = Nj = exp^logn,-)")) is
not visited before n if j is large enough. Consequently for all but finitelymany n there exists a u0 = uo(n,uj) € Q(exp(logn)a+2e) such that Q(uo,N) C
Q(exp(log n)a+3e) is not visited before n. This proves Theorem 22.10.
Now we consider the density K(N,n) for even larger N. The case N = na
will be investigated and for any 0 < a < 1/2 we prove that K(na,n) has a limit
distribution. In fact we have
THEOREM 22.11 (Revesz, 1989/A). For any 0 < a < 1/2 there exists a
distribution function Ga{x) with Ga@) = 0,Ga(l + 0) = 1 and
limP{K([na],n) < x) = Ga{x) (-00 < x < oo)
for almost all x.
At first we present a few lemmas.
LEMMA 22.12 Let na(logn)-^ < ||x|| < Cna @ < a < 1/2,0 > 0,C > 1).Then
Proof. It is a trivial consequence of Theorem 19.3.
LEMMA 22.13 Let x and y be two points of Z2 such that
na(logn)^ < ||x||, ||y||, ||x - y|| < Cna @ < a < 1/2,0 > 0,C > l).
Then
lim m2{x, y; n) =A 2a)
. B2.22)n—»oo 1 — Q;
Proof. Lemmas 22.6 and 22.12 imply
-
y\ njlm^x; n) + m^y; n)]<
x- y;n)
NON-ACTIVATEDVERSIONwww.avs4you.com
LARGE COVERED BALLS 239
B2.23)= + o f^) .
1 - a V log n J
Similarly
m2{x,y;n) >
mi(x- y\n-qn)[mi(x;qn) + m1(y;gn) - m2{x,y;qn)\
Consequently
H^^). B2.20
B2.23) and B2.24) together imply B2.22).The next lemma is an extension of Lemma 22.13.
LEMMA 22.14 . Let xu x2,..., xk (k = 1,2,...) be a sequence in Z2 such that
na(logn)~^ < \\xi - Xj\\, \\xi\\ < Cna
where 0 < a < 1/2,/? > 0, C > 1,1 < i < j < k. Then
Proof. Lemmas 22.5 and 22.12 imply
O (mk < -.
V
(f1Og"
By induction we obtain
NON-ACTIVATEDVERSIONwww.avs4you.com
240 CHAPTER 22
Similarly
rrik > —
j-
Consequently
and by induction we obtain Lemma 22.14.
Proof of Theorem 22.11. Let A(s, n) be the set of all possible s-tuples(xi, x2,...,xs) of Q(na) with the property
na„-.„,„-. -,„_
logn
Then
(X!,«a x.)€A(t,n)
and by Lemma 22.14 we obtain
Urn E((tf(n",n)n = A - 2a)']& (l - (l - j) 2a
Consequently we have Theorem 22.11 with a distribution Ga(-) satisfying
(z) = A - 2o)' ) 2]lY1
Remark 2. The above equality easily implies that Ga(l — e) < 1 for any0 < a < 1/2 and e > 0 and in turn limsupn_>00 K(na,n) = 1 a.s. for any0 < a < 1/2.
NON-ACTIVATEDVERSIONwww.avs4you.com
LARGE COVERED BALLS 241
It is natural to ask the following
Question. Is it true for any 0 < a < 1/2 and e > 0 that
Ga(e) > 0?
22.3 Completely covered balls in Zd
Theorem 22.1 describes the area of the largest disc around the origin covered
by the random walk {Sk,k < n}. In Zd (d > 3) the analogous problem is
clearly meaningless since the largest covered ball around the origin is finite with
probability 1. However, one can investigate in any dimension the radius of the
largest ball (not surely around the origin) covered by the random walk in time
n. Formally speaking let
Q{u,N) = {x:xeZd, \\x - u\\ < N}
and R*(n) = R*(n,d) be the largest integer for which there exists a random
vector u = u(n) G Zd such that Q(u,R*(n)) is covered by the random walk in
time n, i.e.
?(x, n) > 1 for any x G Q{u, R*{n)).
Then we formulate our
THEOREM 22.12 (Revesz, 1989/C). For any e > 0 and d> 3 we have
(logn)^+e eUUC{R*{n)) B2.26)
ande
e LLC(R*(n)). B2.27)
At first we present a few lemmas.
LEMMA 22.15 For any d>3 there exists a positive constant C& such that
P{Sn = x for some n} = P{J{x) = 1} = ^J^ {R - oo)
where R = \\x\\ and
jix\ =i° *
' | 1 o
?(*,n) =0/or every n = 0,1,2,...,otherwise.
NON-ACTIVATEDVERSIONwww.avs4you.com
242 CHAPTER 22
Remark 1. A somewhat weaker version of Lemma 22.15 was found by Erdos -
Taylor A960/A).It is worthwhile to mention the following analogue of the above lemma for
Wiener process.
LEMMA 22.16 (Knight, 1981, p. 103). For any d>Z
( r \ d~2
P{W(t) e Q{u,r) for some t} = I —
J
provided \\u\\ = R > r.
Put a = r, b = R, c = oo, then this lemma is a simple consequence of Lemma
17.1.
Proof of Lemma 22.15. Clearly
P{Sn = x} = f^P{Sk = x,S^x,j = 0,1,2,...,k- n_fc = 0}
and
n=0 fc=0n=0
n=0
Since by Lemma 16.5
,y = 0,1,2,... ,fc - l}P{5n_fc = 0}
= x,S,?x,j = 0,1,2,...,k-l}.k=0
? P{Sn = x} = (Kd + o(l))R2-d (R - oo)n=0
B2.28)
and by Lemma 16.2
we obtain
f;P{5n=0}<oo,n=0
P{J(x) = 1} = ?p{5fc = x,Sj ? x,j = 0,1,2,...,k - 1} =
k=0
Hence we have the Lemma.
o{l))R2-d.
Remark 1. For a more exact form of B2.28) see Spitzer A964), Pi (p. 308)and Problem 5 (p. 339).
NON-ACTIVATEDVERSIONwww.avs4you.com
LARGE COVERED BALLS 243
LEMMA 22.17 For any 0 < a < 1 and L > 0 t/iere eztsts a sequence xi, x2,...,
Xy o/ t/ie points of Zd such that
L< \\xi\\ < L + l (i = l,2,...,T),
||xt-x;|| >La {i,j = 1,2,...,T;i^j),
where K = K(d) is a positive constant depending on d only.
Proof is trivial.
LEMMA 22.18 Let
Dk+l-Dka = (Xi = (Xi(k) =
where
and define T = Tk and x\,X2,...,xt as in Lemma 22.17. Then for any L bigenough we have
P{Q@, L) is covered eventually}
i, X2,.. •, xj are covered eventually} < e~^T~x\
Proof. Define the sequence au aj,...,^ (k = 1,2,...) by
Dk+i_Dk+i-i°* =
Dk+i _ x(t = 2,3,...,fc).
Assuming that
Dk _ Dk-i Dk -I0 < e < —rr-r: <
we have
0 < c^ < a2 < ... < ak < 1 (k = 1,2,...)and
(o,-+1 - ai){d - 1) < Oi{d - 2) (i = 1,2,..., k)where a-k+i = 1-
Let xtl, x,-2,..., x,T be an arbitrary permutation of the sequence x1? x2,..., xy.
Consider the consecutive distances
NON-ACTIVATEDVERSIONwww.avs4you.com
244 CHAPTER 22
Assume that among these distances /i resp. l-i resp. ... /* are lying between Lai
and La* resp. La2 and L� resp., ..., Lak and 2L.
Then (by Lemma 22.15) the probability that the random walk visits the points
1,-j, Xi2,..., XiT in this given order is less than or equal to
Taking into consideration that the number of those j's (l < j' < T) for whichLai < ||x, — i,|| < Lai+l (where s is a fixed element of the sequence A,2,..., T))is less than or equal to
we get
P{xi, X2,... ,xt are covered eventually }
v A
I
= ELi=l
<
if L is big enough and we have Lemma 22.18.
In the same way as we proved Lemma 22.18 we can prove the following:
LEMMA 22.19 For any L big enough and u 6 Zd we have
P{Q(u,L) is covered eventually} < C*Lde-{T-l)
where
D* -1 _ d-l
k is an arbitrary positive integer and C* = C*(d) is a positive constant.
Proof of B2.26). Let L = [(logn)'] with
c al I ( Dk-l Yle 6 +{
NON-ACTIVATEDVERSIONwww.avs4you.com
LARGE COVERED BALLS 245
Then T > (log n)^ with some t/j > I and we obtain our statement B2.26) ob-
observing that
lim 0k = j-t—-7 7-7—-r + e.fc-00
*
(d - 2) - e(d - 1)In order to prove B2.27) we present a few further lemmas.
LEMMA 22.20 There exists a constant K > 0 such that
,n) > 0} > (^) R2~d R=\\x\\
if n > KR2 where C& is the constant of Lemma 22.15.
Proof. Lemma 16.5 easily implies
KR2 ts
n=l
if K is big enough, which, together with the method of proof of Lemma 22.15
implies Lemma 22.20.
Let
and define
T\ = n H
tjji - inf{k : k> TUSn = Sk} - tx,
T2 = Tx+Vl + IOog"K^1!.V»a = inf {fc : fc > t2, Sn = Sk} - r2,...
Clearly, with a positive probability (depending on n) t/^ is not defined. However,we have
LEMMA 22.21 There exists a constant C > 0 such that
P{ max tb{ <
Proof. Clearly for any C > 0 there exists a constant 0 < p = p(C) < 1 such
that
F{||5T1 - Sn|| < C(logn)^T} > p > 0.
NON-ACTIVATEDVERSIONwww.avs4you.com
246 CHAPTER 22
Since
\\STl - Sn\\ <
and 0i, 02, • • • are i.i.d.r.v.'s we have
- Yp\ >n""!
for a suitable C > 0. Hence we have Lemma 22.21.
LEMMA 22.22 Let
A = An = {max 0,;< C(logn)^} and B(n) =
Then among the events B(n) only finitely many will not occur with probability 1.
Proof is trivial.
Proof of B2.27). Let x e Zd satisfying the inequality
Then (by Lemma 22.20)
;n)^T])-?(z,n)=0}<l-
Hence the conditional probability (given An) that x is not covered is less than or
equal to
l - ^(log»)<-<'-*>-1><'-'>)L < exp (-^(logn)^�)) .
Consequently the conditional probability that there exists a point x G
Q((\ogn)(d~l) l~e,5n;(i) being not covered is less than or equal to
NON-ACTIVATEDVERSIONwww.avs4you.com
LARGE COVERED BALLS 247
This fact together with Lemma 22.22 proves B2.27).Theorem 22.12 tells us that the path of the random walk in its first n steps
covers relatively big balls. It is natural to ask where these big covered balls are
located in Zd. For example we might ask about the radius of the largest covered
ball within Q(u,n). In fact let p(n) be the largest integer for which there exists
an r.v. u = u(n) 6 Rd such that ||u|| < n and Q(u,p(n)) is covered by the
random walk eventually, i.e.
J(x) = 1 for any x G Q(u,p(n)).
As a trivial consequence of Theorems 22.12 and 18.5 we obtain
THEOREM 22.13 For any n big enough, d>3 and e > 0 we have
(logn)^T"e < p{n) < (logn)?^+e a.s.
Remark 2. In Section 22.1, for d = 2 we investigated the area of the largestcircle around the origin covered by the random walk {Sk,k < n}. In the presentSection the volume of the largest covered ball in Zd d > 3 (not surely around the
origin) was investigated. Clearly the latter question can be studied in the case
d = 2 too but nothing is known about this problem.
Remark 3. Applying Theorem 7.1 (cf. also the proof of Theorem 19.6) one can
obtain that
Cd{\ogn)l'deLLC{Rd{n)}
for a suitable C^ > 0. It is a somewhat weaker statement than B2.27) but in
some sense it says a bit more than B2.27). In fact we can obtain (by Theorem
7.1) that:
for any n big enough there exists a positive random integer un < n — Kd log n
such that the path {SVn, SUn+l,..., Sl/n+Kd\ogn} covers a ball of volume Cd(\og nI/*.Remark 3 suggests to ask about the connection between the location of the
favourite value and the largest covered ball. For example, we can propose the
following
Question. Let Q(u(n),R*(n)) be the largest covered ball and let xn G Zd be a
favourite value, i.e ?(xn,n) — f(n). Is it true that
xn e Q{u{n), R*{n)) i.o. a.s.?
NON-ACTIVATEDVERSIONwww.avs4you.com
248 CHAPTER 22
22.4 Once more on Z2
The results of Section 22.3 suggest the question whether the largest completelycovered disc is located around the origin or there exists a larger completelycovered disc located somewhere else. The answer to this question is unknown.
However, we present the following:
THEOREM 22.14 For any e > 0 we have
lim sup J2 I{x,n) = l,
where
^'^"{o if ?{x,n)=0.
Proof. By Remark 2 of Section 22.2, for any e > 0 and 6 > 0 there exists a
0 < 7 = "i(e,6) < 1 such that
>-7 (n = l,2,...). B2.29)
Consider the points n0 = 0, nx = [c^\ > n2 =
- By B2.29)
where, f 1 if Sj; = x at least for one nk < j <
k^ ' '
| 0 if 5,-^z for every n* < j' < nk+l.
Then choosing C big enough we obtain the Theorem.
To see the meaning of the above theorem it is worthwhile to compare it with
Theorems 22.9, 22.11 and Remark 2 of Section 22.2.
NON-ACTIVATEDVERSIONwww.avs4you.com
Chapter 23
Speed of escape
Theorem 18.5 on the rate of escape suggests that the sphere {x : ||z|| = R} is
crossed about R times by the random walk if R is big enough and d > 3. In fact
if ||5n|| = y/n for every n then HxeZ(R) f(z,°°) = O(R) where
Z{R) = {x : x e Zd,\\\x\\ - R\ < l}.
Introduce also the following notations:
j(\-f° if ?(*,") =0 for every n = 0,1,2,...,* '
[1 otherwise
and
z€Z{R)
i.e. J(x) = 1 if x G Zd is visited by the random walk and 0(R) is the number
of points of Z(R) visited eventually by the random walk. On the behaviour of
0(R) I have the following:
Conjecture 1. For any d > 3 there exists a distribution function H(x) = Hd{x)for which H@) =0 and
lim P (^ < x\ = H{x) (-00 < x < oo).—»°o
Unfortunately I cannot settle this conjecture but the analogous question for
a Wiener process can be solved. In order to present the corresponding theorem
we introduce the following notations.
Let W(t) = {W^t), W2{t),... ,Wd{t)} {d > 3) be a Wiener process.
249
NON-ACTIVATEDVERSIONwww.avs4you.com
250 CHAPTER 23
Definition. W(t) is crossing the sphere {x : \\x\\ = R} 6 = 0(R) times if 6(R) is
the largest integer for which there exists a random sequence 0 < o^ = ax (R) <
ft = ft(JK) < a2 = a2{R) < ft = 02{R)... < a9 = ae{R) < 0e = 06{R) < oo
such that
\\W(t)\\ <R if t < au
\\W{ai)\\=RtR-l<\\W{t)\\<R+l if ^ <*</?!,
\\W{0l)\\=R-lorR + l,\\W(t)\\^R if ft < t < a,,
\\W{a2)\\ = R, R-l< \\W{t)\\ <R+1 if a2 < t < 02,
\\W{02)\\ =R- lor 12 + 1, \\W(t)\\ ?R if ft<*<as,...
\\W{a,)\\ =R, R-K\\W{t)\\ <R+l if a0<t<00,
\\W@9)\\ =R + l\\W(t)\\>R if t>09.
F(R))~l will be called the speed of escape in R.
THEOREM 23.1 (Revesz, 1989/A).
<•} — •-' «->
The proof is based on the following:
LEMMA 23.1
P{6(R) = k} = A(l - A)*� (A: = 1,2,...)
w/icrc
nand
where B{R,t) = {inf{s : s > t, \\W{s)\\ = R-l} > mf{s : s > t, \\W{s)\\ = R+l}.
Remark 1. Note that the last formula for p(R) comes from Lemma 17.1. ByLemma 22.16
2
= P{\\W(t + s)\\<R ioTsomes>0\\\W(t)\\ = R + l}.
NON-ACTIVATEDVERSIONwww.avs4you.com
SPEED OF ESCAPE 251
Proof. Clearly we have
P{6{R) = 1} = A,
P{0(R) = 2}/ / R \d-t\ / R \d~2 (1- (-= )+P(R)[~?—7i PWi1
d-2-\
where q(R) = 1 — p{R)- Similarly
PF(r) = k)
i-a\ *-i
\k-\=AA-A)'
Hence we have Lemma 23.1.
Observe that
IA"
p(R) 1 - A - jL)--*~
p(/2)(rf - 2)
Since p(R) -> 1/2 (i2 -» oo) we have
Lemma 23.1 together with B3.1) easily implies Theorem 23.1.
Studying the properties of the process {6(R),R > 0} the following questionnaturally arises: does a sequence 0 < Ri < i22 < • • • exist for which
lim Rn = oo and 0(i2j = 1 i = 1,2,...?n»oo
The answer to this question is affirmative. In fact we prove a much strongertheorem. In order to formulate this theorem we introduce the following
Definition. Let ijj(R) be the largest integer for which there exists a positiveinteger u = u(R) < R such that
6(k) = 1 for any u < k < u + tp(R).
It is natural to say that the speed of escape in the interval (u,u + i/j(R)) is
maximal.
NON-ACTIVATEDVERSIONwww.avs4you.com
252 CHAPTER 23
THEOREM 23.2,.
. log log Rtp{R) > t.o. a.s.
log 2
Proof. Let
f[R)-log 2
'
A(R) = {6{k) = 1 for every R < k < R + f{R)}
and
C(R,t) = {\\W(t)\\ = R + f(R) + l}.Then
?{A{R)} = JI P{B(R+j,t) | \\W{t)\\ = R+j);=o
x(l -P{\\W(t + s)\\ < R + f(R) for somes > 0 | C(R,t)})1 d-2
~
logi? 2R
and
logii!log 2
if log log A/log 2 < S = o(R). In the case 5 > O(R) the events A(R) and
+ 5) are asymptotically independent. Hence
and for any e > 0 if n is big enough we have
hS))<(^-=| (loglognJ(l+?)2
iEl 5_r!?t!1L| V /' log 2 I
which implies Theorem 23.2 by Borel - Cantelli lemma.
Conjecture 2.
hm :—7——= a.s.
tf-oo log log it log 2
Theorem 23.2 clearly implies that 0(R) = 1 i.o. a.s. It is natural to ask: how bigcan 6(R) be? An answer to this question is
NON-ACTIVATEDVERSIONwww.avs4you.com
SPEED OF ESCAPE 253
THEOREM 23.3 For any e > 0 we have
6{R) < 2(d-2)-1{l+e)R\ogR a.s.
if R is big enough and
9{R) > A - e)R log log log R i.o. a.s.
Since this result is far from the best possible one and the proof is trivial we
omit it.
Remark 2. Conjecture 1 suggests that 0(R) ~ R. Instead of investigating the
path up to oo consider it only up to px = min{fc : k > 0, Sk = 0}. Takinginto account that P{pi = oo} > 0 if d > 3 we obtain T,xez{R) ?{x,Pi) ~ R
with positive probability. Investigating the case d = 1 by Theorem 9.7 we get
E?«€*(*) Z{x,Pi) = E(?(*!,Pi) + ?(-*!,Pi)) = 2 for any R G Z\ We may ask
about the analogous question in the case d = 2. By Lemma 22.1 we obtain
Lemma 17.1 suggests that the probability of returning to the origin from Z(R-l)before visiting Z(R) is O(R~l(log R)~l). Hence we conjecture that T,xez{R) €{xiPi~ O(R); for example,
= 0(R) as
\x€Z(R) )
for any d > 2.
NON-ACTIVATEDVERSIONwww.avs4you.com
Chapter 24
A few further problems
24.1 On the Dirichlet problem
Let U be an open, convex domain in R2 which is bounded by a simple closed
curve A. Suppose that a continuous real function / is given on A. Then the
Dirichlet problem requires us to find a function u = u(x, y) which
(i) is continuous on U + A,
(ii) agrees with / on A,
(iii) satisfies the Laplace equation
d*u d2u
dy2_
A probabilistic solution of this problem is the following. Let {W(t),t > 0}be a Wiener process on R2 and for any z ?U define Wz(t) = W{t) + z. Further,let az be the first exit time of Wz(i) from U, i.e.
oz = mm{t : Wz{t) <E A} (zG U).
Then we have
THEOREM 24.1 The function
u(z) = Ef(cz)
is the solution of the Dirichlet problem.
255
NON-ACTIVATEDVERSIONwww.avs4you.com
256 CHAPTER 24
The proof is very simple and is omitted. The reader can find a very nice
presentation in Lamperti A977), Chapter 9.6.
Here we present a discrete analogue of Theorem 24.1. Instead of an open,
convex domain U we consider a sequence Ur (r = 1,2,...) of domains defined as
follows.
Consider the following sequences of integers
2 < ... < anr,
c2,. ..,&nr_i <
satisfying the conditions
A) 6t+1 < ct, ct+1 > bi (i =1,2,. /., nr- 2),
B) Oi+1-
a,- > or, c,- b{ > ar,
with some a > 0 and nr = 2,3, Now let
Condition A) implies that Ur is connected. Condition B) has only some minor
technical meaning. Let Ar be the boundary of Ur and define a "continuous"
function /,.(•) on the integer grid of Ar, where by continuity we mean:
For any e > 0 there exists a 6 > 0 such that \f{zi) — f(z2)\ < ? if ||zi — z2\\ < 6r
where zuz2 € ArZ2.Now we consider a random walk {Sn;n = 0,1,2,...} on Z2 and for any
ze{Ur + Ar)Z2 we define
SP = Sn + z (n = 0,l,2,...).
Let az be the first exit time of S,W from Ur, i.e.
az = min{n : 5^ G Ar}.
We wish to prove that
«(*) = E/EW)is the solution of the discrete Dirichlet problem, meaning that
(i) u is "continuous" on (Ur + Ar)Z2, i.e. for any e > 0 there exists a 6 > 0
such that if zuz2 € (Ur + Ar)Z2 and ||zi-Z2|| < ^»", then |uBx) -u(z2)\ < ?,
(ii) u agrees with / on ArZ2,
NON-ACTIVATEDVERSIONwww.avs4you.com
A FEW FURTHER PROBLEMS 257
(iii) u satisfies the Laplace equation, i.e.
(u(x + 1, y) - 2u(x, y) + u{x - 1, y))+ (u{x, y + 1) - 2u(x, y) + u(x, y
- 1)) = 0
whenever (x, y), (x + 1, y), (x, y + 1), (x - 1, y), (x, y- 1) <E (tfr + Ar)Z2.
(ii) is trivial, (iii) follows from the trivial observation that
u(x, y) = -(u(x + 1, y) + u(x - 1, y) + u(x, y + 1) + u(x, y- 1))
4
if (x, y) satisfies the condition of (iii).In order to prove (i) we present a simple
LEMMA 24.1 For any e > 0 there exists a 6 > 0 suc/i t/iat if
zeUrZ\ qeArZ2, \\z-q\\<6r
then
Consequently\Ef(S{M)M)-f(q)\<e\
Proof is simple and is omitted.
In order to prove (i) we have to investigate two cases:
(a) zuz2eUrZ\
(/?) one of zi, z2 is an element of ArZ2 and the other one is an element of UrZ2.
In case (/?) our statements immediately follow from Lemma 23.1. In case (a)assume that oZl < aZ2 and observe that
Since S(*2V*J = S^tM2)^'^(<rs^){OMi)) applying Lemma 23.1 with q = S
and z = S^(aZl) we obtain (i).
Remark 1. Having the above result on the solution of the discrete Dirichlet
problem, one can get a concrete solution by Monte Carlo method. In fact to getthe value of u(-, •) in a point zq = (x0, yo) & Ur observe the random walk starting
NON-ACTIVATEDVERSIONwww.avs4you.com
258 CHAPTER 24
in zq till the exit time aZn and repeat this experiment n times. Then by the law
of large numbers
lim n'1 ? /(sH(ffJ) = u(x0, y0) a.s. B4.1)»oo
*—¦"n—»oo
where Si,S2>... are independent copies of a random walk. Hence the average
in B4.1) is a good approximation of the discrete Dirichlet problem if n is big
enough. A solution of the continuous Dirichlet problem in some zo or in a few
fix points can be obtained by choosing r big enough and the length of the stepsof the random walk small enough comparing to the underlying domain.
24.2 DLA model
Let A\ C Ai C ... be a sequence of random subsets of Z2 defined as follows:
Ax consists of the origin, i.e. A\ — {0},
A2 = A\ + y2 where y2 is an element of the boundary of At
obtained by the following chance mechanism. A particle is released at oo and
performs a random walk on Z2. Then y2 is the position where the random walk
first hits the boundary of Ax.The boundary of a set A C Z2 is defined as
dA = {y : y E Z2 and y is adjacent to some site in A, but y ? A}.
For example, dA, = {@,1), A,0), (-1,0), @, -1)}.Having defined An,An+i is defined as An+i = An + yn+i where yn+i is the
position where the random walk starting from oo first hits dAn.In the above definition the meaning of "released at oo" is not very clear.
Instead we can say: let
Rn = inf{r : r > 0, An C Q{r) = {x : \\x\\ < r}}.
Then instead of starting from infinity the particle might start its random walk
from (R^,0) (say). It is easy to see that the particle goes round the origin before
it hits An (a.s. for all but finitely many n). This means that the distribution of
the hitting point will be the same as in case of a particle released at oo.
Many papers are devoted to studying this model, called Diffusion Limited
Aggregation (DLA). The reason for the interest in this model can be explained
NON-ACTIVATEDVERSIONwww.avs4you.com
A FEW FURTHER PROBLEMS 259
by the fact that simulations show that it mimicks several physical phenomenawell.
The most interesting concrete problem is to investigate the behaviour of the
"radius"
rn = max{||x|| : x? An}.
Trivially rn > (m/ttI/2 and it is very likely that rn is much bigger than this trivial
lower bound. Only a negative result is known saying that rn is not very big. In
fact we have
THEOREM 24.2 (Kesten, 1987). There exists a constant C > 0 such that
limsup n~2/3rn < C a.s.
n—»oo
The proof of Kesten is based on estimates of the hitting probability of dAn.He proved that there exists a C > 0 such that for any y 6 dAn we have
P{yn+i = y}< Cr-W. B4.2)
In order to get a lower estimate of rn we should get a lower estimate of the
probability in B4.2) at least for some y G dAn. Auer A989) studied the questionof how one can get the lower bounds of the hitting probabilities of some pointsof the boundaries of certain sets (not necessarily formed by a DLA model). He
investigated the following sets:
Bi = {(-r,0), (-r + 1,0),..., (r - 1,0), (r,0)},B2 = Bx + {@, -r), @, -r + 1),..., @, r - 1), @, r)},B3 = {x = (xi, x2) : \xi\ + \x2\ = r}.
Consider the point y = (r,0). Then the probability that the particle coming from
infinity first hits y among the points of dBi(i — 1,2,3) is larger than or equal to
Cr'1'2 if i = 1,2
and
(gr)-1'3 if i =
with some C > 0.
NON-ACTIVATEDVERSIONwww.avs4you.com
260 CHAPTER 24
24.3 Percolation
Consider Z2 and assume that each bond (edge) is "open" with probability p and
"closed" with probability 1 —
p. All bonds are independent of each other. An
open path is a path on Z2 all of whose edges are open.
One of the main problems of the percolation theory is to find the probability0(p) of the existence of an infinite open path. Kesten A980) proved that
0(P)\>O if p>l/2.
The value 1/2 is called the critical value of the bond percolation in Z2.An analogous problem is the so-called site percolation. In site percolation the
sites of Z2 are independently open with probability p and closed with probabilityq = 1 —
p. Similarly as in the case of the bond percolation a path of Z2 is called
open if all its sites are open and we ask the probability @*(p) of the existence of
an infinite open path. The critical value of the site percolation in Z2 is unknown,but T6th A985) proved
0*(p) = 0 if p < xAQ ~ 0,503478
where x0 is the root lying between 0 and 1 of the polynomial
We call the attention of the reader to the recent survey of Kesten A988) on
percolation theory.
NON-ACTIVATEDVERSIONwww.avs4you.com
And God said, "Let there be lights in the
firmament of the heavens to separate the
day from the night; and let them be for
signs and for seasons and for days and
years."
The First Book of Moses
III. RANDOM WALK IN RANDOM
ENVIRONMENTNON-ACTIVATEDVERSION
www.avs4you.com
Notations
1. ? = {... ,E_2,E_i,Eo,Ei,E2,...} is a sequence of i.i.d.r.v.'s satisfying0 < Ei <1- 0 with some 0 < 0 < 1/2 called environment.
2.
3.
j^n^T^Pe},^?,?}; see Introduction.
4.
5.
¦ *) =
D(b) =
0
1
= 1 + Ux
if 6 = a,
if 6 = a + 1,
if b>a + 2,
6.
D{0,n - 1)_
?)@,n)~
i.e.
e° + exp(-(rB_i - Tn_2)) + expHT,,.! - rn_3)) +
+ exp(-(r,_i - To)) = D{n)e-T"-> (n = 1,2,...).
263
NON-ACTIVATEDVERSIONwww.avs4you.com
264 III RANDOM WALK IN RANDOM ENVIRONMENT
7.
(n = 1,2,...). Caution: D{n) = D{0,n); however, D(-n) ? D(-n,O).
8. I(t) is the inverse function of D(n), i.e.
I[t) = k if D{k) <t<D{k + 1),
I{-t) =ki( D{-k) < t < D(-k - 1) (t > 1; k = 1,2,...).
9. Rq, Ru ... is a random walk in random environment (RWIRE); see Intro-
Introduction.
10. p(a, 6, c); see Lemma 27.1.
NON-ACTIVATEDVERSIONwww.avs4you.com
Chapter 25
Introduction
The sequence {Sn} of Part I was considered as a mathematical model of the
linear Brownian motion. In fact it is a model of the linear Brownian motion in
a homogeneous (non-random) environment.
We meet new difficulties when the environment is non-homogeneous. It is the
case, for example, when the motion of a particle in a magnetic field is investigated.In this case we consider a random environment instead of a deterministic one.
This situation can be described by different mathematical models.
At first we formulate only a special case of our model. It is given in the
following two steps:
Step 1. (The Lord creates the universe). The Lord visits all integers of the real
line and tosses a coin when visiting t (t = 0, ±1,±2,...). During the first six
days He creates a random sequence
? ={..., ?_2, E-i, Eo, Ei, ?2, • • •}
where ?, is Head or Tail according the result of the experiment made in t.
Step 2. (The life of the universe after the Sixth Day). Having the sequence
{..., E-2, E-i, Eo, Ei, E2,...} the Lord puts a particle in the origin and givesthe command: if you are located in t and Ei is Head then go to the left with
probability 3/4 and to the right with probability 1/4, if Ei is Tail then go to
the left with probability 1/4 and to the right with probability 3/4. Creating the
universe and giving this order to the particle "God rested from all his work which
he had done in creation" forever.
The general form of our model can be described as follows:
Step 1. (The Lord creates the universe). Having a sequence
265
NON-ACTIVATEDVERSIONwww.avs4you.com
266 CHAPTER 25
? = {..., E-2, E-i,E0, Ei, E2,...} of i.i.d.r.v.'s with distribution
<x} = F{x), F@) = 0,
the Lord creates a realization ? of the above sequence. (The random sequence
{..., E-2, E-\, EQ, Ei, E2, •. •} and a realization of it will be denoted by the same
letter ?.) This realization is called a random environment (RE).
Step 2. (The life of the universe after the Sixth Day). Having an RE ? the
Lord lets a particle make a random walk starting from the origin and going one
step to the right resp. to the left with probability Eq resp. \ — Eq. If the particleis located at x = i (after n steps) then the particle moves one step to the rightresp. to the left with probability Ei resp. 1 — Ei. That is, we define the random
walk Rq, Ri, ..., by iZo = 0 and
=« + l | Rn =
1 - Pf (i^n+i = i! - 1 | Rn = i,Rn-i, Rn-2, ...,Ri) = Ei. B5.1)The sequence {J?n} is called a random walk in RE (RWIRE).
A more mathematical description of this model is the following. Let {ftl5 /i,Pi}be a probability space and let
{...E-t = E.2{ui),E-i = E-i{ui),E0 = E0{ui),Ei = El{ul),E2 =
[ux e nx) be a sequence of i.i.d.r.v.'s with Pi(#i < x) = F{x){F@) = l-F(l) =
o).Further, let {fi2, 72} be the measurable space of the sequences u2 = {el5 e2,...}
where e, = 1 or e, = — l(t = 1,2,...) and J2 is the natural er-algebra. Define
the r.v.'s YX,Y2,... on fi2 by Yi{u2) = e:t(t = 1,2,...) and let Rq = 0,J?n =
Yi + Y2 + ¦- • + Yn(n = 1,2,...). Then we construct a probability measure P
on the measurable space {ft = f^ x ft2, 7 = J\ x J2} as follows: for any givenU\ G fti we define a measure PWl = P?(Wl) = Pf on J2 satisfying B5.1). (ClearlyB5.1) uniquely defines Pf on J2.) Having the measures P?((Jl)(u;1 G fti) and Pione can define the measure P on J the natural way.
Our aim is to study the properties of the sequence {Rn}- In this study we
meet two types of questions.
(i) Question of the Lord. The Lord knows ux, i.e. the sequence ?; or in other
words, He knows the measure Pf and asks about the behaviour of the
particle in the future, i.e. He asks about the properties of the sequence
{Rn} given ?.
NON-ACTIVATEDVERSIONwww.avs4you.com
INTRODUCTION 267
(ii) Question of the physicist. The physicist does not know wi. Perhaps he hassome information on F, i.e. he knows something on P^ He also wants to
predict the location of the particle after n steps, i.e. also wants to describethe properties of the sequence {Rn}-
A typical answer to the first type of question is a theorem of the followingtype:
THEOREM 25.1 There exist two sequences of Immeasurable functions /M =
A) < fi2) = /?(?) such that
P < m*x\Rk\ < fW a.s. (P,) B5.2)
for all but finitely many n, i.e.
p?{fi1]{?) < max \Rk\ < fi2){?)for all but finitely many n) = 1.
Since the physicist does not know the environment S he will not be satisfied with
an inequality like B5.2). However, he wants to prove an inequality like
THEOREM 25.2 There exist two deterministic sequences a^ < a^ such that
c^ </^ </<2> < a<2> a.s. (Pl) B5.3)
for all but finitely many n.
Having inequalities B5.2) and B5.3) the physicist gets the following answer to
his question:
THEOREM 25.3 There exist two deterministic sequences a^ < a^ such that
cH><mBx\Rk\<ag) a.s. (P) B5.4)
for all but finitely many n. Equivalently
P{aJ' < max \Rk\ < a™ for all but finitely many n}
aJ,1) < max \Rk\ < a^] for all but finitely many n) = 1} = 1.
Remark 1. The exact forms of Theorems 25.1, 25.2 and 25.3 are given in
Theorems 27.6, 27.8 and 27.9 where the exact forms of ato,fM,ag\fW are
given.
Remark 2. In the special case when
Pi(?o = 1/2) = F(l/2 + 0) - F(l/2) = 1,
the RWIRE problem reduces to the simple symmetric random walk problem.
NON-ACTIVATEDVERSIONwww.avs4you.com
Chapter 26
In the first six days
In this chapter we study what might have happened during the creation of the
universe, i.e. the possible properties of the sequence ? are investigated.The following conditions will be assumed:
(C.I) there exists a 0 < 0 < 1/2 such that P(/? < Eo < 1 - /?) = 1,
(C2)
EiV0 = f°° xdP^Vo < x) = I' "log-—-dF{x) = 0y-oo Jp x
where F{x) = P^Eq < x),V0 = \ogU0 and Uo = (l - Eo)/Eo,
(C.3)
0 < a2 = ExV02 = (log dF{x) < oo.
Jp \ x /
Remark 1. In case of a simple symmetric random walk (i.e. Pi(i?o = 1/2) = 1)we have Pi(C/0 = 1) = Pi(V0 = 0) = 1 and consequently (C.l) and (C.2) are
satisfied; however, (C.3) is not satisfied since E^2 = a2 = 0. We also mention
that if (C.I) and (C.2) hold and E^2 = a2 = 0 then ^(Eo = 1/2) = 1.
Remark 2. Most of the following theorems remain true replacing (C.l) by a
much weaker condition or omitting it. Here we are not interested in this type of
generalizations.
LEMMA 26.1
HmsupTn = limsupT_n = -liminf Tn = — liminf T_n ;= oo a.s. (Pi).— - n—*oo n—^00
B6.1)n—»oo
269
NON-ACTIVATEDVERSIONwww.avs4you.com
270 CHAPTER 26
If we assume (C.I) and (C.3) but instead of (C.2) we assume that EiV0 = m^0.Then
lim Tn = lim T_n = (sign m)oo a.5. (P^ B6.2)n—»oo n-
Tn = n + V2 + ••• + Vn,T_n = V_i + V_2 + ••• + V_n,V;- = log #,-,#,• =
A - ?,)/?, and To = 0.
Proof. B6.1) is a trivial consequence of the LIL of Hartmann and Wintner (cf.Section 4.4), B6.2) follows from the strong law of large numbers.
LEMMA 26.2
lim Din) = oo a.5. (P^ B6.3)n—»oo
' ' v '
(cf. Notation 5.).
Proof. Since
D{n) = l + Ul + UlU2 + --- + UlU2...Un-l=e° + eT>+eT* + --- + eT"->, B6.4)
B6.3) follows from Lemma 26.1.
By B6.4) we have
exp( max Tk) < D(n) < nexp( max Tk) B6.5)0<Jk<n-l
; — V ; —
^VO<Jk<n-l ; V ;
and the LIL implies
LEMMA 26.3 For any e > 0 and for any p = 1,2,... we have
max Tk < A + e:)erBnloglognI/2a.s. (Px) for all but finitely many n. B6.6)
max Tk > A - er)crBnloglognI/2 i.o. a.s. (Px), B6.7)
max Tk < n^^lognloglogn-'-logpn)� i.o. a.s. (Pj, B6.8)
max Tk > nlsJcsfl
a.s. (Pi) for all but finitely many n. B6.9)
By B6.5) we also get
D(n) < exp{(l + er)crBnloglognI/2}a.5. (P^ for all but finitely many n, B6.10)
NON-ACTIVATEDVERSIONwww.avs4you.com
IN THE FIRST SIX DAYS 271
D{n) >exp{(l-e)erBnloglognI/2} i.o. a.s. (Pi), B6.11)
D{n) <exp{n1/2(lognloglogn--logpn)�} i.o. a.s. (P^, B6.12)
D{n) > exp{n1/2(lognloglogn---(logpnI+e)-1}a.s. (Pi) for all but finitely many n. B6.13)
Replacing the maxi<*<n by max_n<jk<_i the inequalities B6.6) - B6.9) remain
true. Replacing D(n) by D(—n) in B6.10) - B6.13) they remain true as theyart'
-S^LpL =a a.s. (Px), B6.14)v2nloglogn
liminf maxlog D^ J\oglogn = ott/Vs a.s. (Pj, B6.15)
n—»oo 0<Jk<n y/n
D*{n)>l, B6.16)
D*{n) < n i.o. a.s. (Pi). B6.17)
Proof. Inequalities B6.6) - B6.13) are clear as they are. The following simpleanalogue of B6.5),
-^min_i(rn_i- Tk)) < D*{n) < nexp(-^min_i(rn_i
- Tk)), B6.18)
implies B6.16) and B6.17).In order to get B6.14) and B6.15) approximate the process {7^,0 < k < oo}
by a Wiener process {erW(t),0 < t < oo}. By Theorem 10.2 the process
-m\nT(W(T)-W(t))is identical in distribution to the process {|W^(OI^ ^ 0}- Hence the LIL and the
Other LIL imply B6.14) and B6.15).
LEMMA 26.4 For any e > 0 and for any p = 1,2,... we have
W < (log |?j log log |i| - - - logp_1 |i|(logp |i|I+eJa.s. (Px) if \t\ is big enough, B6.19)
I{t) > (log jt| log log |t| - - - logp |t|J t.o.a.s. (Px), B6.20)
•¦*"¦ (Pi)' B6-21)
/@>•19iglOglll a-5" (Pi) * \t\is big enough. B6.22)2G log \t\
NON-ACTIVATEDVERSIONwww.avs4you.com
272 CHAPTER 26
Proof. B6.19), B6.20), B6.21) resp. B6.22) follows from B6.13), B6.12),B6.11) resp. B6.10).
LEMMA 26.5
l) (*>1), B6.23)
D{-I{-t)) < t < D{-I{-t) - 1) {t > 1), B6.24)
D{n + 1) = D{n) + UXU2 ¦Un = D{n) + UnJpj~A (n = 1,2,...), B6.25)
+ ^5,) < D(W) < t, B6.26)
D{n + 1) < ^D(n) + 1, B6.27)
I{Xt + 1) > 7@ + 1 if A > i B6.28)
where 0 is the constant of (C.l).
Proof. B6.23), B6.24), B6.25) follow immediately from the definitions. B6.23)and B6.25) combined imply
This, in turn, implies B6.26). B6.27) follows from (C.l). B6.28) follows from
B6.23) and B6.27).
NON-ACTIVATEDVERSIONwww.avs4you.com
Chapter 27
After the sixth day
27.1 The recurrence theorem of Solomon
THEOREM 27.1 (Solomon, 1975). Assuming conditions (C.I), (C.2), (C.3)we have
P{Rn = 0 t.O.} = Pl{P?{i*n = 0 t.O.} = 1} = 1.
Assuming (C.I), (C.3) and EiV0 ^ 0 we have
Remark 1. The statement of the above Theorem can be formulated as follows:
with probability 1 (Pi) the Lord creates such an environment in which the recur-
recurrence theorem is true, i.e. the particle returns to the origin i.o. with probability1 (Pf). Before the proof of Theorem 27.1 we present an analogue of Lemma 3.1.
LEMMA 27.1 Let
p(a, b, c) = P?{min{j : j > m, R, = a} < min{j : j > m, R3 = c} \ Sm = b}
(a < b < c), i.e. p(a,b,c) = p(a,b,c,?) is the probability that a particle startingfrom b hits a before c given the environment S. Then
Especially
p(O,l,n) = 1-—-r- and p(O,n-l,n) = —
D(n)yK ' ' ;
D*(n)
273
NON-ACTIVATEDVERSIONwww.avs4you.com
274 CHAPTER 27
Proof. Clearly, we have
p(a,a,c)=l, p(a,c,c)=0,
p(a, 6, c) = Ebp{a, b + 1, c) + A - Eb)p{a, b-l,c).
Consequently,
p{a, b + 1, c) - p(a, 6, c) = —^"t(p(o, 6> c) ~ P(a>6 ~ 1.c))-
By iteration we get
p{a, b + 1, c) - p(a, 6, c) = UbUb-i • • • Ua+l{p(a, a + 1, c) - p(a, a, c))= UbUb.x • • • C7a+I(p(a, a + 1, c) - 1). B7.1)
Adding the above equations for 6 = a, a + 1,..., c — 1 we get
-1 = p(a, c, c) - p(a, a, c) = D(a, c)(p(a, a + 1, c) - 1),
*a>a+l>c)rl-Dh)- B7-2)
Hence B7.1) and B7.2) imply
p(a, b + 1, c) — p(a, 6, c) = --=-t rEW6_i • • • Ua+l.V{a,c)
Adding these equations we obtain
p(a, b + 1, c) - 1 = p{a, b + 1, c) - p(a, a, c)= t^-tA + Ua+l + Ua+lUa+2 + ¦¦¦ + Ua+lUa+2 ¦¦¦Ub)
_
D{a,b + 1)D{a,c)
Hence we have the Lemma.
Consequence 1.
p(ll» p@,l,*;?) =nlim A-^=1) = 1, B7.3)
P{limp(-n,-l,0;?) = 0} = 1. B7.4)
NON-ACTIVATEDVERSIONwww.avs4you.com
AFTER THE SIXTH DAY 275
B7.3) follows from Lemma 27.1 and B6.3). In order to see B7.4) observe
(-n,-l) 1
and apply B6.13) for D(-n).The following lemma is a trivial analogue of Lemma 3.2; the proof will be
omitted.
LEMMA 27.2 For any -oo < a < b < oo we have
P{liminf Rn = a} = PllimsupiZn = 6} = 0.n—»oo
Proof of Theorem 27.1. Assume that Rx = 1, say. Then by Lemma 27.2 the
particle returnes to 0 or it is going to +oo before returning. However, by B7.3)for any e > 0 there exists an n0 = no(e,?) such that p@,1, n) = 1 —
—^> 1 — e
if n > no. Consequently the probability that the particle returns to 0 is largerthan 1 — e for any e > 0 which proves the Theorem.
27.2 Guess how far the particle is going away
in an RE
Introduce the following notations:
M+(n) = max J?t,V ' 0<ifc<n
M~(n) = — min Rk,
M{n) = max{M+(n),M-(n)} = max \Rk\,Osifc^fl
Po = 0,
Pi = min{ifc : k > 0, Rk = 0},
= mm{k : k > ph Rk = 0},
?(n) = max?(A:,n),
u{n) = #{«: 0 < « < n - 1, RPi+i = 1}.
Observe that ?@, pn) = n.
NON-ACTIVATEDVERSIONwww.avs4you.com
276 CHAPTER 27
Our aim is to study the behaviour of M(n). Especially in this section a
reasonable guess will be given.Consider the simple environment when
Note that conditions (C.I), (C.2) and (C.3) are satisfied. Note also that in the
environment ? = {..., 3/4,1/4,3/4,1/4,3/4,...} the behaviour of the random
walk is the same as that of the simple symmetric random walk. For example, it
is trivial to prove that
lim sup bnM(n) = 1 a.s.
n—*oo
if ? is the given environment and bn = Bnloglogn)~1/2.One can guess that since environment {..., 3/4,1/4,3/4,1/4,...} is nearly
the typical one, M(n) will be practically n1/2 in most environments. This way
of thinking is not correct because we know that in a typical environment there
are long blocks containing mostly 3/4's and long blocks containing mostly 1/4's.Assume that in our environment
max Tk = n1/2 and - min T-k = n1/2l<Jfe<n l<*<
which is a typical situation. Then by B6.10), B6.11) and Lemma 27.1 we have
p@,1, n) = 1 - -pJ" 1 - exp(-n1/2)
and
p(-n,-1,0) =
This means that the particle will return to the origin exp(n1/2) times before
arriving n or — n. Hence to arrive n requires at least exp(n1/2) steps. Conversely,in n steps the particle cannot go farther than (lognJ.
This way of thinking is due to Sinai A982). He was the first one who realized
that having high peaks and deep valleys in the environment, for the particle it
takes a long time to go through. Clearly high peak means that T(k) is a bigpositive number for k > 0 resp. it is a big negative number for k < 0 while the
meaning of the deep valley is just the opposite.
NON-ACTIVATEDVERSIONwww.avs4you.com
AFTER THE SIXTH DAY 277
27.3 A prediction of the Lord
LEMMA 27.3 For any environment ? we have
TE{u{n) = k} = fyEfr - E0)n-k, B7.5)
limsup,o rj^f0} u/2=l «.-• (P.) B7.6)
where un is defined in Section 27.2.
Proof is trivial.
LEMMA 27.4 For any environment ? and k = 1,2,... we have
) <k\yn} = (p@, L ( )"
W <*}= ?(
Proof is trivial.
Now we prove our
THEOREM 27.2 For any environment ? we have
<M+(pn) </(n(lognI+e) a.s. (P?), B7.9)< M-{pn) < 7(-n(lognI+e) a.s. (P?), B7.10)
max{/(n(logn)-1-e),/(-n(logn)-1-e)} < M(pn)< max{/(n(lognI+e),/(-n(lognI+e)} a.s. (P?) B7.11)
for all but finitely many n.
Proof. By Lemma 27.4 and B6.26) we have
/
1--
NON-ACTIVATEDVERSIONwww.avs4you.com
278 CHAPTER 27
\ n
< 1- 1-E0Qn
-n(logn)
E0Qn
-(logn)1+e
where
and N = N{n) = I Qn(log nI+e) .
D*{N)Let nk = 2k. Then by the Borel - Cantelli lemma we get (cf. B6.16))
a.s.
for all but finitely many k. If nk < n < nk+1 we have
M+(pn) < M+(pnk+i) <
< / fn(lognI+eJ a.s. (P^) for all but finitely many n
Hence we have the upper part of B7.9).Now we turn to the proof of the other inequality of B7.9).By Lemma 27.4 and B6.26) we have
= 1-
Hence we have B7.9) by the Borel - Cantelli lemma.
The proof of B7.10) is identical. B7.11) is a trivial consequence of B7.9) and
B7.10).In order to get some estimates of M(n) (resp. M+(n),M~(n)) the Lord is
interested to estimate pn or equivalently ?@, n). To study this problem in a more
general form we present a few results describing the behaviour of the local time
?(z,n).LEMMA 27.5 For any integer k = 1,2,... and any environment ? we have
Eo1-
D(k)
E0(l-
if 1 = 0,
Consequently
D{k)D
Eo
IK1-^)'1*'-1'2'-"
D(k)
L-\
(?=1,2,...).
NON-ACTIVATEDVERSIONwww.avs4you.com
AFTER THE SIXTH DAY 279
Proof. Clearly we have
,Pi) = 0} = 1 - Eo + E0p{0,l,k)
D(k)In case / = 1,2,... we get
:,Pl) = 1} = E0(l -p(O,l,k))(l - Ek)p(O,
y=o \ ^ )
E0(l-Ek) / l-^A'�Ek) ( 1-
I>(ik)I>*(ik) V !>*(*) /
A trivial calculation gives
LEMMA 27.6 (Csorgo - Horvath - Revesz, 1987). For any k = 1,2,...
B7'12)
/or any A < - log(l - ^J). Especially
^l-e*(l-2A)Observe that
and2AeA
~2 l/ 0<A<1/2-
Proof. As an example we prove B7.14). By Lemma 27.5 we have
+-;i-^*)
{
NON-ACTIVATEDVERSIONwww.avs4you.com
280 CHAPTER 27
Remark 1. B7.12) implies that: for any e > 0
mk > -J— exp((l - e)crBJkloglogikI/2) i.o. a.s.
and
mk< l--^-exv{-{l-e)o{2k\og\ogkI/2) i.o. a.s.
Compare these inequalities and (9.6).
LEMMA 27.7 For any A; = 1,2,... and any environment ? we have
nmU(k,Pn)-nmk
<1
-°° \ y/nak J
limsup ^P^^ = i a.5. (p?). B7.18)n-*oo ak y/2n log log n
Proof is trivial.
Now we give a somewhat deeper consequence of B7.14).
LEMMA 27.8 (Csorgo - Horvath - Revesz, 1987). For any
and any k = 1,2,..., we have
X2E? exp(A(?(A:, Pl) - mk)) = 1 + —a\ + X3ek
where5
>k\ ^ and A is a positive constant.
Proof. By Taylor expansion we get
eA D*{k) 1
D(k)h(X) + (D(k)h(X)J + n(D(k)h(X)K)
NON-ACTIVATEDVERSIONwww.avs4you.com
AFTER THE SIXTH DAY 281
with |0| < 1, | r) |< 1, where
Consequently
f_ x(f_if A is big enough. Hence by B7.14) we have
E?expA?(*,pi) - (l + Amt + A2
A2(Z?(A:)J - i <A\\\*(D(k)L
(D(k) \l-E
Multiplying the above inequality by
A2 A3exp(-Amfc) = 1 - Xmk + —m\+r)—rn\-
one gets the Lemma.
LEMMA 27.9 Let
0<x<ak fmin|v^,-v^logfl--^Then for any k = 1,2,... and n = 1,2,... we have
,P») " nmk\ > xy/n} < 2exp
Proof. Apply Lemma 27.8 with
A = xn�/2^2 and A
Then we get
,pn) - nmk))2)
<2
and we have the Lemma.
This last inequality gives a very sharp result for ?(fc, pn) when k is not too
big. In cases where k can be very big it is worthwhile to give another consequence
of Lemma 27.6. In fact we prove
NON-ACTIVATEDVERSIONwww.avs4you.com
CHAPTER 27
LEMMA 27.10 For any K> 0 there exists aC = C{K) > 0 such that
C\ognD*{k)}<n-K B7.19)
Proof. Let A = Xk = ±$j. Then by B7.15) and B7.16) we get
+ CDt{k)\ogn},pn) > expBAnmfc + XCD*{k) logn)}, Pl))n exp(-2Anmfc - \CD* [k) log n)exp
1 exp (n("
which proves B7.19).A very similar result is the following:
LEMMA 27.11 For any CY > J (cf. (C.l)) we have
< exp (-{k = 1,2,. ..;n = 1,2,...).
Proof. Let A = Afc = ^*y. Then by B7.15) and B7.16) we get
> exp -J J< exp (-
exp f-
\E?(exp(\Z(k,pn)))
E?(expA?(A:,pi))|
2AeA
B7.20)
< exp ( -
NON-ACTIVATEDVERSIONwww.avs4you.com
AFTER THE SIXTH DAY 283
Hence we have B7.20).An analogue result describes the behaviour of ?(A:,pi) when A: is a big positive
number.
LEMMA 27.12 There exist positive constants C and C\ such that
??{Z{k,Pi) > CxD*{k)\ogk | ?(*,Pi) > 0} < Cxk-2 B7.21)
and
i) > 0} < CAT2. B7.22)
Proof. Let fj.k be the number of negative excursions away from k between 0 and
P\. Clearly, we have
Pi) > 0} = p@,k- l,*)(l - p@,* - I,*)I�
Consequently
P?{nk > L | e(*,Pi) > 0} = A - (D'ik))-1I-. B7.23)
Hence
0} < P,{m* < k^D^k) \ Z(k,Pl) > 0}= 1 - A - [Dt{k))-1)k"D'^ < Ck~2
and we have B7.22). In order to prove B7.21) observe that for any 0 < 6 < e
there exists a C2 = C2{6) > 0 such that
where ^fc = 1 - Ek - 6. Hence by B7.23) we have
log A: | ?(*,pi) > 0}l?*(*) log*,m* > ^*e(*,Pi) I ?(*,Pi) > 0}!l?*(*) log^M* < Ekt(k,p!) | e(*,Pi) > 0}
log A: | ?(*,pi) > 0}
f) P?{»k < Ekl | ?(*,pi) = i}Pf(e(*,pi) > 0}<=C!D#(Jk) log Jk
which proves B7.21).In the following lemma we investigate the probability of the event that ?(k, pn)
is very small.
NON-ACTIVATEDVERSIONwww.avs4you.com
284 CHAPTER 27
LEMMA 27.13
- Ek p@, k - 1, k)
with some constant C > 0.
Proof. Let
l if,
max RPi+j>k,0 otherwise,
S = Srt = fo + ?i + --- + ?n-i.
Then
pas = i} = ?o(i-p(o,i
and by the Bernstein - inequality (Theorem 2.3)
Let 1 < ti < i2 < ... < is < n be the sequence of those t's for which
max Rp+i > k
and let
i/,- = e(fc,p4i+i) - ?(*,*,.) (j = 1,2,...,5),
i.e. Vj is the number of excursions away from k between pii and p,-y+i. Further,let i/;~ resp. i/t be the number of the corresponding negative resp. positiveexcursions. Then
!/,- = !/+ +1/7,
T>e{vJ =m} = (l-q)m-lq, q = (l -
and using again the Bernstein - inequality we obtain
NON-ACTIVATEDVERSIONwww.avs4you.com
AFTER THE SIXTH DAY 285
Hence
~
< -nmk J• • • + v~ < -nmk, S < -npj• • • + v~ < -nmk, S > -u
< Cexp (-^j + Pf |i/f + i/f + • • • + I/." < -nmk, S
Since Vj > vj we have the Lemma.
Now we give an upper bound for pn.
THEOREM 27.3 For any e > 0 and for all but finitely many n we have
Clogn j^ ?>*(fc) a.5. (Pf) B7.24)Jk=-/(-n(logn)»+«) Jk=-/(n(logn)»+«)
where C is a big enough positive constant.
Proof. By Theorem 27.2 we have
/(n(tofn)»+«)
Jk=-/(-n(logn)»+«)
Lemma 27.10 and the Borel - Cantelli lemma imply
I(n(logn)l+*)
Clogn ^ D*{k).k=0 Jk=O Jk=O
Analogous inequality can be obtained for negative fc's. Hence we have B7.24).A somewhat weaker but simpler upper bound of pn is given in the following:
THEOREM 27.4 For any e > 0 and for all but finitely many n we have
/(n(lofn)»+')
Pn < n{\ognJ+e ? mk a.s. (Pf).Jk=-/(-n(logn)»+«)
NON-ACTIVATEDVERSIONwww.avs4you.com
286 CHAPTER 27
Proof. By B7.12), B6.23) and (C.l) of Chapter 26 for any 0 < k < /(n(lognI+e),we have
Hence)
Clogn J2 D*{k)<n{\ognJ+2< ? mk.
k=0 Jk=O
Since analogous inequality can be obtained for negative fc's we have the Theorem.
A lower bound for pn is the following:
THEOREM 27.5
pn>-maxmk a.s. (P?) B7.25)4 cA
where A = An = {k : 0 < D(k) < k^},k < C/12 and f3 is defined in (C.l) ofChapter 26.
Proof. Since
pn > max
B7.25) follows by Lemma 27.13.
Remark 2. Remark 1 easily implies that
lim maxmjk = oo a.s. (Pi).
Hence B7.25) is much stronger than the trivial inequality pn > In.
Clearly having the upper bound B7.11) of M(pn) and the lower bound B7.25)of pn we can obtain an upper bound of M(n). Similarly having the lower bound
B7.11) of M(pn) and the upper bound B7.24) of pn a lower bound of M(n) can
be obtained. In fact we have
THEOREM 27.6 Let
ft{n) = max{/(n(lognI+<),/(-n(lognI+«)},f;{n) = max{/(n(logn)-1-e),/(-n(logn)-1-e)},
/(n(logn)'+<)
g(n) = n(lognJ+< ?
hln) = —
v '4
Jk=-/(-n(lognI+«)
NON-ACTIVATEDVERSIONwww.avs4you.com
AFTER THE SIXTH DAY 287
Then for all but finitely many n
f:(g-lH)<M(n)<f?(h-l(n)) a.s. (P,) B7.26)
where g~l{-) resp. h~l(-) are the inverse functions of g() resp. h(-).
Proof. Theorems 27.2 and 27.4 combined imply
/; < M(pn) < M(</(n)),
which, in turn, implies the lower inequality of B7.26). Similarly by Theorems
27.2 and 27.5 we get
f?(n)>M(pn)>M(h(n))and we have the upper inequality of B7.26).
Remark 3. Note that knowing the environment ? the lower and upper bounds
of B7.26) can be evaluated.
27.4 A prediction of the physicist
Having Theorem 27.2 and Lemma 26.4 the physicist can say
7(n(lognI+e) < (lognloglogn-.-logp^nOogpnI^J B7.27)and the analogue inequalities are true for M~(pn) and M(pn). Theorem 27.2
and Lemma 26.4 also suggest that B7.27) and the corresponding inequalities for
M~(pn) and M(pn) are the best possible ones. It is really so. In fact we have
THEOREM 27.7 (Deheuvels - Revesz, 1986). For any e > 0 and p = 1,2,...we have
M+{pn) < (lognloglogn-.-logp^nOogpnI^J a.s. (P)if n is big enough, B7.28)
M > (lognloglogn--logp_1nlogpnJ i.o. a.s. (P), B7.29)
~? °S n
as (P) ^ n n big enough. B7.31)log3 n
The same inequalities hold for M~(pn) and M(pn).
NON-ACTIVATEDVERSIONwww.avs4you.com
288 CHAPTER 27
Proof. B7.27) gives the proofs of B7.28) and B7.31). Since by Theorem 27.2
for all but finitely many n
M+{pn)>I{n{\ogn)-1") a.s. (P?)
and by B6.20)
/(n(logn)-1-') > ((logn - (l + 2er) log2 n) log2 n- • • logp+1 nJ> (lognlogjfi- • -logpTiJ i.o. a.s. (Pi)
we have B7.29). Similarly, by Theorem 27.2
M+(pn)<I(n(\ognI+g) a.s. (P?),
and by B6.21)
hence we get B7.30). Clearly, the physicist is more interested in the behaviour
of M+(n),M~(n),M{n) than those of M+(/>„), M" (/>„), M(/>n). Since pn > In
by B7.28) and B7.30) we have
THEOREM 27.8 (Deheuvels - Revesz, 1986). For any e > 0 and p = 1,2,...we have
M+(n) <(lognloglogn---logp_1n(logpnI+'J a.s. (P)if n is big enough, B7.32)
and
The same inequalities hold for M~(n) and M(n).
To get a lower bound for M+(n),M~(n) and M(n) is not so easy. However, as
a consequence of Theorem 27.6 we prove
THEOREM 27.9 (Deheuvels - Revesz, 1986). For any e > 0 we have
for any e > 0 and for all but finitely many n. The same inequality holds forAf-(n) andM{n).
NON-ACTIVATEDVERSIONwww.avs4you.com
AFTER THE SIXTH DAY 289
Proof. Let+
0+(n)=n(logn)a+* ? mk.
Jk=O
Then by Condition (C.I), B7.12), B6.19) and B6.6)
9+(n) < ^n(lognJ+< ? e
Jk=O
< -—^n(lognJ+€/(n(lognI+e) max
x exp((l + 2e)crB(lognJ(log2 n)
<exp(logn(loglognI+2e). B7.35)
It can be shown similarly that for any e > 0
g{n) <exp(logn(loglognI+e) a.s. (Px) B7.36)
for all but finitely many n. Consequently
^) a" (Pl) B7'37)
if n is big enough. Hence by B6.22)
B7.26) and B7.38) combined imply the Theorem.
NON-ACTIVATEDVERSIONwww.avs4you.com
Chapter 28
What can a physicist say about the
local time ?@,n)?
28.1 Two further lemmas on the environment
In this section we study a few further properties of the environment ?. These
results are simple consequences of the corresponding results of Part I.
LEMMA 28.1 For any 0 < e < 1 and 0 < 6 < e/2 there exists a random
sequence of integers 0 < nx = n1(u;1; e, 6) < n2 = n2(oJi; ?,6) < ... such that
Tn. <-A - e)ob~l and max Tj < n\/2{log nk)~s ¦ B8.1)
where bn = b(n) = Bnloglogn)�/2. Consequently by B6.5) and B6.18)
^2~5) and
B8.2)
Proof. B8.1) follows from E.11) and Invariance Principle 2 of Section 6.3.
LEMMA 28.2 There exist a random sequence 0 < nx = n1(u;1) < n2 =
n2(oJi) < ... and two constants Cx > 0, C2 > 0 such that
Tnk > C2b~l and
( \1/2max {Ti - Tj) < Cl —-*
. B8.3)<»<><nit
" -
Vloglogn/v '
291
NON-ACTIVATEDVERSIONwww.avs4you.com
292 CHAPTER 28
Consequently by B6.5) and B6.18)
D{nk) > exp (C7b-l) and
max D*{j) < exp \CX \—~^ ) I . B8.4)o<.<y<nt
wy *
\ \ loglognjk / / '
Proof. B8.3) is a simple consequence of Theorem 10.5 and Invariance Principle2.
28.2 On the local time f@, n)Since ?{Q,pn) = n Theorem 27.4 and B7.36) imply
THEOREM 28.1 For any e > 0 we have
>n) < ?@,exp(logn(loglognI+e))
i.e.
('°^.) a,. (P) B80!)
for all but finitely many N.
Now we prove that B8.5) is nearly the best possible result. In fact we have
THEOREM 28.2 For any e > 0
'°^,) ,-....... (P). B8.6)
Proof. Define the random sequence {Nk} as follows: let Nk be the largestinteger for which
where nk is the random sequence of Lemma 28.1. Then by B7.9)
M+(pNk) > I(Nk(\ogNk)-ll+<M) > /(iV^logiV,)-1-') + 2 > nk a.s. (P)
for all but finitely many k, i.e. ?{nk,PNk) > 0. That is to say, there exists a
0 <j =j{k) < Nk such that ?{nk, (Pj,pj+l)) = Unk,Pj+i)-^{nk,Pj) > 0. Hence
by B7.22)P?{Z{nk,{Pj,pj+l)) < n-k2D*{nk)} < Cn~k\
NON-ACTIVATEDVERSIONwww.avs4you.com
WHAT CAN A PHYSICIST SAY ABO UT f@, n) ? 293
and by B8.2) and the Borel - Cantelli lemma
?(»*,(P/,Pi+i)) > nl2D*{nk) > exp((l - e)ab~lk) a.s. (P)
for all but finitely many k (where j = j{k)). Consequently
pNk > P,-+i-
Pj > ?K, (p,-,P,-+i)) > exp((l - e)ab-lk) a.s. (P) B8.7)
for all k big enough. By B6.23) and B8.2)
Vfc)-A+<) < D(I(Nk(\ogNky(l+e)) + 1) < ?>(nfc) < exp(ni/2
i.e.
n*>^log2iVfc(loglogiVfc)W. B8.8)
B8.7) and B8.8) combined imply for any 6* < 1 and for all but finitely many k
PNk>exp(\ogNk{\og\ogNkY') a.s. (P). B8.9)
B8.9) in turn implies Theorem 28.2.
Theorems 28.1 and 28.2 have shown how small ?@, N) can be. Essentiallywe found that ?@,iV) can be as small as N1/Xo*XogN. In the next two theorems
we investigate the question of how big ?@, N) can be. In fact we prove
THEOREM 28.3 There exists aC = C(f3) > 0 such that
^@, N) > exp ( (l - ^-j^J log Nj i.o. a.s. (P)
where C is defined in condition (C.I).
Proof. By B7.12) we have
Eo D(j) D(j)B8-10)
Hence by Lemma 27.10 for any K > 0 there exists a C = C(K) > 0 such that
n) > CnD'U)} <n~K {j = 1,2... ,n = 1,2,...). B8.11)
Define the random sequence {Nk} as follows: let Nk be the smallest positiveinteger for which
>nfc B8.12)
NON-ACTIVATEDVERSIONwww.avs4you.com
294 CHAPTER 28
where {nk} is the random sequence of Lemma 28.2. Observe that by B8.4) and
B6.23) for all but finitely many k
exp (C2b-nl) < D(nk) < D(I(Nk(\ogNk)l+<)) < Nk(\ogNk)l+e a.s. (Px).
Hence
L|^^ a.s. (PO, B8.13)3 Nk
and by B7.9) for all but finitely many k
M+(pNk)<I(Nk(\ogNkI+''2)<nk a.s. (P).
ConsequentlytU,Psk) = 0 if j>nk. B8.14)
By B8.10), Lemma 27.10 and B8.13) for any K > 0 there exists a C = C(K) > 0
such that
)*(j) J < nkNkK < ±^-^Nk-«. B8.15)
Hence by the Borel - Cantelli lemma, B8.13), B8.14), B8.15) and B8.4) we get
< CNknkexV L fp-^-H <7*^
x
VloglognJ J~
C\7*^J C\ \og3Nk
C%r log2 Nk (d \ogNk\Nexv{)
for all but finitely many k. Since similar inequality can be obtained for the sum
Ef=i Z{-J,PNk) and for pNk = E^_oo ZU>PNk) we have
if k is big enough, which implies Theorem 28.3.
Looking through the above proofs of Theorems 28.2 and 28.3 one can realize
that somewhat stronger results were proved than stated. In fact we have proved
NON-ACTIVATEDVERSIONwww.avs4you.com
WHAT CAN A PHYSICIST SAY ABOUT ?@, n) ? 295
THEOREM 28.4 For almost all environment ? and for all e > 0 and C bigenough there exist two random sequences of positive integers
nx = n\{?,e) < n2 = n2(?,e)... and
mx = mx(?,C) < m2 = m2(?,C) < ...
such that
?@,nfc) <exp
and
«„,«,)> op ((l-j^Jlogm,).Remark 1. Theorems 28.1 - 28.3 are, as we call them, theorems of the physicist.However, Theorem 28.4 can be considered as a theorem of the Lord. Knowingthe environment ? the Lord can find the time-points where ?@, •) will be very
big or very small while the physicist can only say that there are infinitely many
points where ?@, •) takes very big resp. very small values but he does not know
the location of these points.In the last theorem of this chapter we prove that ?@, n) cannot be very close
to n, i.e. Theorem 28.4 is not far from the best possible one.
THEOREM 28.5 For any C > 0 we have
?@, n) < exp((l - 6n) log n) a.s. (P)
for all but finitely many n where
Proof. Introduce the following notations:
M+{pi,Pj+i) = max Rk G = 1,2,...),
tfj*(N) = max{n : 0 < n < N,T{n) < -cri;1},n),Af+(p,-_1,p,-) > x}.
Note that by Theorem 5.8 (especially Example 3) and by Strong Invariance
Principle 2 we obtain
max Tk < e(b(*l;*(N)))-1 a.s. (P) B8.16)
NON-ACTIVATEDVERSIONwww.avs4you.com
296 CHAPTER 28
for any e > 0 and for all but finitely many n. Hence by Lemma 27.1, B6.5) and
B8.16)
> nexp(
Consequently if C is big enough then for any n = 1,2,... we have
and by the Borel - Cantelli lemma
J",n) a.a. (P) B8.17)
for all but finitely many n. Applying Lemma 27.1 and the definition of ij)*(N)we obtain
-O < exp (-|Hence by Lemma 27.1, B8.17) and B8.18) we get
n
^j
(JV) - l)))�) . B8.19)
Applying Theorem 5.3 we obtain
for all but finitely many N. B8.19) and B8.20) imply that with some C > 0
n > ffi^le(o,n)exp(^F@*(iV)))-1) > e@,n)exp^j a.s. (P)
for all but finitely many n where
Hence we have Theorem 28.5.
Remark 2. Since
there is an essential gap between the statements of Theorems 28.3 and 28.5.
NON-ACTIVATEDVERSIONwww.avs4you.com
Chapter 29
On the favourite value of the RWIRE
In this chapter we investigate the properties of the sequence ?(n) = max* ?(k, n).A trivial result can be obtained as a
Consequence of B7.32). For any e > 0 we have
a.s. (P). B9.1)
We also get
Consequence of B7.33).
B9.2)
It looks obvious that much stronger results than those of B9.1) and B9.2) should
exist. In fact we prove (Theorem 29.1) that (under some extra condition on ?)
^0 a.s. (P).pn—*oo Tl
THEOREM 29.1 (Revesz, 1988). Assume that
Pl(Ei = p) = ^{Ei = 1 - p) = \ @ < p < 1/2). B9.3)
Then there exists a constant g = g(p) > 0 such that
limsupn-1^(n) > g(p) a.s. (P).
297
NON-ACTIVATEDVERSIONwww.avs4you.com
298 CHAPTER 29
Remark 1. Very likely Theorem 29.1 remains true replacing condition B9.3)by the usual conditions (C.I), (C.2), (C.3). Note that B9.3) implies (C.I), (C.2)and (C.3).
At first we introduce a few notations. Let N be a positive integer,
and define the random variables on
p+{N) = min{k :k>0,Tk = AiV},
p-(N) = min{k :k>0,T.k = -AiV},
/4A = -minlr* : 0 < k < pt{N)},= max{r_fc : 0 < k < p
For the sake of simplicity from now on we assume that n% > /i^. Continue the
definitions as follows:
aN = max{Jfc : 0 < k < pf{N),Tk = -/iNA},_
_
f maxjifc : 0 < k < aN, Tk + fxNA = AiV} if such a k exists,
max{k : k < 0, -Tk + fxN A = AiV} otherwise,
fc :aN <k<pt{N),Tk + nNA = AiV},
LN(j) = L(-A(fMN-j),(T^,r^)) =#{k:T^<k< r+,Tk = -/inA +JA},= max{ry - T{ : r^ < i < j < aN},= max{r, - Ty :
and on fl:
Fn = min{k : k > 0, Rk = aN},GN = min{k : k > 0, Rk =
Hff = rain{k : k > F^, Rk = ^ or riJ} ~ Fn-
For the sake of simplicity from now on we assume that t# < 0.
The above notations can be seen in Fig. 1, where instead of the process Tkthe process
Tk =
is shown.
_
f Tk if k > 0,~
\ -Tk if k < 0
NON-ACTIVATEDVERSIONwww.avs4you.com
ON THE FAVOURITE VALUE OF THE RWIRE 299
Figure 1
Now we present a few simple lemmas.
LEMMA 29.1 There exists an absolute constant 6 @ < 0 < 1) such that
where
Proof. Consequence 1 of Section 13.3 easily implies that
Ti{LN(j) < 6/2 + Aj + 2,y = 0,1,..., N - 1} B9-4)is larger than an absolute positive constant independent from N. It is easy to
see that
?-,/-(*)<-} B9.5)is larger than an absolute positive constant independent from N (cf. Consequence4 of Section 10.2) and the events involved in B9.4) and B9.5) are asymptoticallyindependent as N —> oo. Hence we have Lemma 29.1.
Let N = Nk(?) be a sequence of positive integers for which
LN{J) < 6/2 + Aj + 2 (j =0,l,...,iV-l), and A holds
(by Lemma 29.1 for almost all ? there exists such an infinite sequence).
NON-ACTIVATEDVERSIONwww.avs4you.com
300 CHAPTER 29
LEMMA 29.2 For almost all ? and for any e > 0 we have
?@, FN) < exp ({1+
Proof. By Lemma 27.1, B6.5) and the definition of N = Nk we have
M >aN} = E0{l- p@,1, aN)) =°
JU{aN)
> —exp(— max Tk) > —exp(-
and
, HN + Fs) > exp((l -e)N),FN< exp (^ + *)N) , B9.7)
B9.8)
(N = Nk) a.s. (Pf) for all but finitely many k and
= o{GN) a.s. (Pf). B9.9)
< A - Eo) exp(- max Tk) = A -
(cf. the notations in Section 27.2). Hence by the Borel - Cantelli lemma we
easily obtain B9.6). The above two inequalities also imply that more excursions
are required to arrive at p~[(N) than at a#. Hence we get B9.9). The first
inequality of B9.7) and B9.8) can be obtained similarly. In order to prove the
second inequality of B9.7) observe that by B9.8) and B9.9)
^=E^W= E aJ,FN)<(aN-l-pj(N))exp((l + e)-)i=-oo i=r7(N)
a.s. (Pf) for any e > 0 and for all but finitely many k. Hence the lemma is
proved.
NON-ACTIVATEDVERSIONwww.avs4you.com
ON THE FAVOURITE VALUE OF THE RWIRE 301
Introduce the following further notations:
Pi = Pi{<*n) = min{n : n > 0,RpN+n = aN},
h = h{aN) = min{n : n > pi,RFf/+n = as},---
U = ?{j,pn) = ?{j,FN+pn) -
aN-2
= 1+ E *xp{-{Tail-i-Tk))
and
D*{j,N) = {1-pUJ+U*n))-1 = D{j,aN).Observe that
p{j,aN-l,aN)= Dh, as) — Dlj, a^
— 1)= rr:
1~P(J,J+ !
= Uj+lUj+2 ¦ ¦ ¦ ?/„„_! = exp(Taw_i - Tj).
Clearly Lemmas 27.11 and 27.10 can be reformulated as follows:
LEMMA 29.3 For any j < an we have
P. U(j,Pn) > Of^j < exp (--^—) B9.10)
where Cx > 2p~l. Further, for any K > 0 there exists aC = C(K) > 0 such that
Pf \iU,Pn) > 2n1~/°"^(-7^) + CD*(j,N) lognl < n~K. B9.11){ tj D{j,N) J
Proof of Theorem 29.1. In order to simplify the notations from now on we
assume that r^ > 0. The case t^ < 0 can be treated similarly.Let 1/2 + e < ^i A < V>2 < 1 — ? with some e > 0 and introduce the following
notations:
n = [exp(V>2iV)] where N = Nk = Nk{?)and
X{j) = min{fc : r^ < k < aN,Tk = -/
Consider any integer / E (x{4>iN),aN). Then
( <aw(, Ta^i)) Np(ViAiV)
NON-ACTIVATEDVERSIONwww.avs4you.com
302 CHAPTER 29
a.s. (Pf) for all but finitely many k and by B9.10)
<expl-expl n
Consequently by B9.9) and Lemma 29.1
aN-l f)*E «y.A.)<c,»
) i
OtN-1
oo
2+ 2J exp(-/A) = /(A)n a.s. (P) B9.12)
j=o
for all but finitely many k.
Let / e {tn,x{1>iN)). Then by B9.11)
g + CD*(l,N) logn) < n^.
Consequently
() () ,
E e(/,Pn)<2(l-f;aN)n j: fJ^r + ClognED{lN) l=T-
2 ^naNexp(-V>iAiV) + C(logn)aNexp ( —) = o(n). B9.13)p \ z /^(VAiV) C(l) (
p
B9.12) and B9.13) combined imply
aN-\
?(J,P»)<2/(A)n a.s. (P) B9.14)
for all but finitely many k. Similarly one can see that
NON-ACTIVATEDVERSIONwww.avs4you.com
ON THE FAVOURITE VALUE OF THE RWIRE 303
Hence by B9.7)
+00
J=-0O T^
Let m = 4/(A)n. Then applying again B9.7) we get
h e)m) > ?(Fn + m) > ?(Fs + pn) > ?(<*#»-FV + pn) = n =
4/(A)
which proves the Theorem.
Note that we have proved a stronger result than Theorem 29.1. In fact we
have
THEOREM 29.2 For almost all environment there exists a sequence of positiveintegers nx = nx{?) < n2 = n2(?) < ... such that
provided that the condition of Theorem 29.1 is fulfilled.
Remark 2. On the connection of Theorems 29.1 and 29.2 the message of Remark
1 of Section 28.2 can be repeated here as well.
Another simple consequence of the proof of Theorem 29.1 is
THEOREM 29.3 Assume that the condition of Theorem 29.1 is fulfilled. Then
there exists an e = e(p) > 0 such that
n—*oo \ n f
On the liminf behaviour of ?(n) we present only a
Conjecture.
lim inf -^—- log log n = 0 a.s. Pn-»oo n
and
liminf—^—(loglognK = oo a.s. P.n—oo n
NON-ACTIVATEDVERSIONwww.avs4you.com
Chapter 30
A few further problems
30.1 Two theorems of Golosov
Theorems 27.8 and 27.9 claim that M{n) ~ (lognJ. As we have already men-
mentioned, this fact was observed first by Sinai A982). The result of Sinai suggestedto Golosov to investigate the limit distributions of the sequences
K=cr2(lognJ
and
*""
cr2(lognJ
(for the definition of a, cf. (C.3) of Chapter 26). In order to study the limit
distributions of R^ and M^ he modified a bit the original model. In fact he
assumed that Eo = 1, i.e. the random walk is concentrated on the positive half-
line. Having this modified model he proved that the limit distributions of the
sequences {i2^} and {M^} existed and he evaluated those. In fact we have
THEOREM 30.1 (Golosov, 1983). For any u > 0
where
lim T{R*n < u} = - /U h2(v)dv C0.1)n—>oo 2./0
l *\ j Bn--zlvj and zn = ±
n=0
and
lim P{M^ <u}= fU hx{v)dv C0.2)
305
NON-ACTIVATEDVERSIONwww.avs4you.com
306 CHAPTER 30
where
n=0
Considering the original model Kesten A983) proved
THEOREM 30.2
2 fulim P{-Rn < u} = - h3(v)dv
n—»oo 7T J—oo
where
Sinai A982) also proved that there exists a sequence of random variables ax, a2,...
defined on Qt such that Rn —
an = o((lognJ) in probability (P). This means that
knowing the environment ? we can evaluate the sequence {an} and having the
sequence {an} we can get a much better estimate of the location of the particleRn than that of Theorem 27.6. Golosov proved a much stronger theorem. His
result claims that Rn —
an has a limit distribution (without any normalising fac-
factor), which means that knowing an the location of the particle can be predictedwith a finite error term with a big probability. The model used by Golosov is
a little bit different from the one discussed up to now. He considered a random
walk on the right half-line only and he assumed that the particle can stay where
it is located. His model can be formulated as follows.
Let ? — (p-i(n),po(n),p1(n),} (n = 0,1,2,...) be a sequence of indepen-independent random three-dimensional vectors whose components are non-negative and
p_i@) = 0, p-i(n) + Po(rc) + Pi(rc) = 1 (n = 0,1,2...). Assume further that
(i) (p-i(n),pj(n)) (n = 1,2...) are identically distributed,
(ii) po(n) (n = 0,1,2,...) are identically distributed,
(iii) the sequences {po{n), n = 0,1,2,...} and {p-i(n)/pj(n), n = 1,2,...} are
independent,
(iv) Elog(p_1(n)/p1(n)) = 0 and 0 < E(log(p.1(n)/p1(n))J = a2 < oo,
(v) E(l - po(n))� < oo and P(po(n) > 0) > 0.
Having the environment ? we define the random walk {-Rn} by Rq =0 and
(p-^i) if 9 =-I,
Y?{Rn+i = i + 0\Rn = i,Rn-U---,Ro} = { Po@ if 0 = 0,
(pi(i) if 0=1-
Then we have
NON-ACTIVATEDVERSIONwww.avs4you.com
A FEW FURTHER PROBLEMS 307
THEOREM 30.3 (Golosov, 1984). There exists a random sequence {an} de-
defined on Qi such that for any— oo < y < oo
lim T{Rn -an<y} = F{y)n—»oo
where the exact form of the distribution function F(y) is unknown.
Remark 1. Clearly Theorems 30.1 and 30.2 can be considered as theorems
of the physicist. However, Theorem 30.3 is a theorem of a mixed type. The
physicist knows about the existence of an but he cannot evaluate it. The Lord
can evaluate an but He cannot use His further information on ?. In fact, He
would like to evaluate the distribution Pf {-Rn —
<*n < */}¦ I* is not clear at all
whether the \\mn-,0O'P?{Rn —
an < y} exists for any given ?.
Remark 2. Theorems 29.1 and 29.2 also suggest that Rn should be close to an.
30.2 Non-nearest-neighbour random walk
The model studied in the first five chapters of this Part is a nearest-neighbourmodel, i.e. the particle moves in one step to one of its neighbours. In the last
model of Golosov the particle keeps its place or moves to one of its neighbours.In a non-nearest-neighbour model the particle can move farther. Such a model
can be formulated as follows.
Let ? = (P-i(rc),Pi(n),P2(rc)} {n — 0,±1,±2,...) be a sequence of inde-
independent, identically distributed three-dimensional random vectors whose com-
components are non-negative and p_i(n) + Pi(n) + p2(n) = 1 (n = 0,±l,±2,...).Then we define a random walk {Rn} by Rq = 0 and
pi[i) if 0 = 1,
P2{i) if 9 = 2.
Studying the properties of {Rn} is much, much harder than in the nearest-
neighbour case. Even the question of the recurrence is very hard. In fact, the
question is to find the necessary and sufficient condition for the distribution
function
which guarantees that
T{Rn = 0 i.o. } = 1. C0.3)
NON-ACTIVATEDVERSIONwww.avs4you.com
308 CHAPTER 30
This question was studied in a more general form by Key A984), who in the
above formulated case obtained the required condition.
THEOREM 30.4 (Key, 1984). Let
<* = Pi@) + P2@) + ((Pl@) + P2@)J + 4p_1@)p2@))J/2Bp_1@))-1
and
(y)Then
P{Rn = 0 i.o.} = 1 if m = 0,
P{ lim Rn = oo} = 1 if m > 0,
P{Jiin Rn = -oo} = 1 if m < 0.vn—»oo
Remark 1. Clearly this Theorem gives the necessary and sufficient condition
of C0.3) if the expectation m exists. If m does not exist then the necessary and
sufficient condition is unknown, just as in the nearest-neighbour case.
Remark 2. The general non-nearest-neighbour case (i.e. when the environment
? is defined by an i.i.d. sequence
where L and R are positive integers) was also investigated by Key. However, he
cannot obtain an explicit condition for C0.1), but he proves a general zero-one
law which implies that
P{Rn = 0 i.o. } = 0 or 1.
His zero-one law was generalized by Andjel A988).
30.3 RWIRE in Zd
The model of the RWIRE can be trivially extended to the multivariate case.
For the sake of simplicity here, we formulate the model in the case d = 2. Let
Un = {UJ}\ulp,U§\ufp) {i,j = 0,±l,±2,...) be an array of i.i.d.r.v.'s with
\? > 0, ?#> + UW + UV + UW = 1. The array ? = {UiJt ij = 0, ±1, ±2,...}
NON-ACTIVATEDVERSIONwww.avs4you.com
A FEW FURTHER PROBLEMS 309
is called a two-dimensional random environment. Having an environment ? a
random walk {Rn, n = 0,1,2,...} can be defined by Rq = 0 and
P{i2n+1 = (», j + l)\Rn = {iJ),Rn-u- --,
P{Rn+1 = (i-hj) I Rn = (i,j),Rn-x,.-.,Ro} = U$\1
= [ij ~l)\Rn = {i,j),Rn-U ¦ ¦ , ^No non-trivial, sufficient condition is known for the recurrence
P{i2n=0 i.O.}=l-
in the case d > 2. Kalikow A981) gave necessary conditions. In fact, he gave a
class of environments where P{-Rn = 0 i.o.} = 0. As a consequence of his result
he proves
THEOREM 30.5 Define the environment ? by
and
Pij^.^.^.^) = {<>2,l>2,c2,d2)} = l-p.
Assume that
p[a\ — C\) I a\ b\ C\ d\\rK ^—r- > max —.-r-.—>T" •
\\ — p)[c2 — a2) \a2 b2 c2 d2)Then
Y{Rn = 0 i.o. } = 0;
moreover
lim RW = oo a.s. (P)
where R^ is the first coordinate of Rn.
Kalikow also proves a zero-one law, i.e. he can prove under some regularityconditions that P{-Rn = 0 i.o.} = 0 or 1. This zero-one law was extended byAndjel A988).
Kalikow also formulated some unsolved problems. Here we quote two of them.
Problem 1. Is every 3-dimensional RWIRE transient?
NON-ACTIVATEDVERSIONwww.avs4you.com
310 CHAPTER 30
Problem 2. LetO<p<l/2 and define the random environment ? by
Is this RWIRE recurrent?
30.4 Non-independent environments
In Chapter 25 we mentioned the magnetic fields as possible applications of the
RWIRE. However, up to now it was assumed that the environment ? consists
of i.i.d.r.v.'s. Clearly the condition of independence does not meet with the
properties of the magnetic fields and most of the possible physical applications.In most cases it can be assumed that the environment is a stationary field. A
lot of papers are devoted to studying the properties of the RWIRE in case of a
stationary environment ?.
In the multivariate case it turns out that having some natural conditions
on the stationary environment ? (which exclude the case of independent en-
environments) one can prove the recurrence and a central limit theorem with a
normalizing factor (lognJ.
30.5 Random walk in random scenery
Let a = (a, = o(i), i = 0, ±1, ±2,...) be a sequence of i.i.d.r.v.'s with
Ea, = 0, Ea? = 1, E(expt<7.) < oo
for some \t\ < t0 (t0 > 0). a is called random scenery. Further, let {5*} (in-(independent from {ak}) be a simple symmetric random walk. Kesten and SpitzerA979) were interested in the sum
fc=0
If the particle has to pay $ a, whenever it visits i, then the amount paid by the
particle during the first n steps of the random walk is n3lAKn. Clearly
+00
fc=-oo
NON-ACTIVATEDVERSIONwww.avs4you.com
A FEW FURTHER PROBLEMS 311
where ?(•, •) is the local time of {Sk}-Studying the sequence {Kn} Kesten and Spitzer are arguing heuristically as
follows: let T/Uo a* = Lh t^ien one can define independent Wiener processes Wxand W2 such that W2(n) should be near enough to Ln and simultaneously ?(k, n)should be near to the local time rji(k,n) of the Wiener process Wi(-). Hence
Kn~n-Z'A ? (W2(k + l)-W2(k))Vl(k,n)k=-oo
+oo
rn[x,n)dWt{x). C0.4)r+oo
J — OO
Since it is not very hard to prove that n 3/4 J+~ rji(x, n)dW2{x) has a limit dis-
distribution, the above heuristic approach suggests that Kn has a limit distribution.
Applying Invariance Principle 2 (Section 6.2) and Theorem 10.1 it is not
hard to get a precise form of C0.4).We note that Kesten and Spitzer investigated a much more general situation
than the above one and they initiated an extended research of random sceneries.
As an example we refer to Bolthausen A989) where a multivariate version of the
above problem is treated.
30.6 Reinforced random walk
Construct a random environment on R1 by the following procedure. Let Rq =
0, P{.Ri = 1} = Y{Ri = -1} = 1/2 and let the weight of each interval (»,» +
1) (t = 0,±1,±2,...) be initially 1 and increased by 1 each time the process
jumps across it, so that its weight at time n is one plus the number of indices
k < n such that (Rk,Rk+1) is either (»,» + 1) or (» + 1,»). Given {Ro = 0, Rx =
»i, ..., Rn = in} Rn+l is either tn + 1 or tn- 1 with probabilities proportional
to the weights at time n of (in,in + 1) and (tn — l,tn) where »i,»2, •••,»'* is a
sequence of integers with |tJ+i — ij\ = 1 {j = l,2,...,n). Hence if i2i = 1, the
weight of [0,1] at time n = 1 is 2. Consequently
= - and -P{R2 = 0 | #i = 1} = ?.3
Similarly
P{i22 = -2 | i2j = -1} = i and P{R2 = 0 | R, = -1} = |.Further, in the case Rx = 1, R2 = 2 the weights of [0,1] and [1,2] at time n = 2
are equal to 2. Hence
1T{R3 = 3\Rl = l,R2 = 2} = 1 - P{i23 l\R1 l,R2 2}
NON-ACTIVATEDVERSIONwww.avs4you.com
312 CHAPTER 30
In the case Ri = 1, R2 = 0 the weight of [0,1] at time n = 2 is equal to 3. Hence
P{i23 = 1 | Ri = 1,R2 = 0} = 1 - P{i23 = -l\R1 = l,R2 = 0} = -.
Similarly
P{i23 = -3\Rl = -l,R2 = -2} = 1 -P{i23 = -l\Ri = -l,R2 = -2} = \and
P{i23 = -1 | Rx = -l,R2 = 0} = 1-P{i23 = 1 | Ri = ~l,R2 = 0} = -.
This model was introduced by Coppersmith and Diaconis (cf. Davis A989))and studied by Davis A989, 1990).
Intuitively it is clear that the random walk generated by this model is "more
recurrent" than the simple symmetric random walk. However, to prove that it
is recurrent is not easy at all. This was done by Davis A989, 1990), who studied
the recurrence in more general models as well.
Note that in this model the random environment is changing in time and
depends on the random walk itself. Situations where the random environment is
changing in time look very natural in different practical models.
NON-ACTIVATEDVERSIONwww.avs4you.com
References
ANDJEL, E. D.
A988) A zero or one law for one-dimensional random walks in random environ-
environments. The Annals of Probability 16, 722-729.
AUER, P.
A989) Some hitting probabilities of random walks on Z2. Preprint.
A990) The circle homogeneously covered by random walk on Z2. Statistics &
Probability Letters 9, 403-407.
AUER, P. - REVESZ, P.
A989) On the relative frequency of points visited by random walk on Z2.
Preprint.
BARTFAI, P.
A966) Die Bestimmung der zu einem wiederkehrenden Prozess gehorendenVerteilungsfunktion aus den mit Fehlern behafteten Daten einer Einzigen Re-
Realisation. Studia Sci. Math. Hung. 1, 161-168.
BASS, R. F. - GRIFFIN, P. S.
A985) The most visited site of Brownian motion and simple random walk. Z.
Wahrscheinlichkeitstheorie verw. Gebiete 70, 417-436.
BERKES, I.
A972) A remark to the law of the iterated logarithm. Studia Sci. Math. Hung.7, 189-197.
BICKEL, P. J. - ROSENBLATT, M.
A973) On some global measures of the deviations of density function estimates.
The Annals of Statistics 1, 1071-1095.
BILLINGSLEY, P.
A968) Convergence of Probability Measures. 3. Wiley, New York.
313
NON-ACTIVATEDVERSIONwww.avs4you.com
314 REFERENCES
BINGHAM, N. H.
A989) The work of A. N. Kolmogorov on strong limit theorems. Theory of
Probability and its Applications 34, 152-164.
BOLTHAUSEN, E.
A978) On the speed of convergence in Strassen's law of the iterated logarithm.The Annals of Probability 6, 668-672.
A989) A central limit theorem for two-dimensional random walk in random
sceneries. The Annals of Probability 17, 108-115.
BOOK, S. A. - SHORE, T. R.
A978) On large intervals in the Csorg6 - Revesz Theorem on Increments of a
Wiener Process. Z. Wahrscheinlichkeitstheorie verw. Gebiete 46, 1-11.
BOREL, E.
A909) Sur les probabilities denombrables et leurs applications arithmetiques.Rendiconti del Circolo Mat. di Palermo 26, 247-271.
BORODIN, A. N.
A982) Distribution of integral functionals of Brownian motion. Zap. Nauchn.
Semin. Leningrad Otd. Mat. Inst. Steklova 119, 13-88.
BROSAMLER, G. A.
. A988) An almost everywhere central limit theorem. Math. Proc. Camb. Phil.
Soc. 104, 561-574.
CHUNG, K. L.
A948) On the maximum partial sums of sequences of independent random vari-
variables. Trans. Am. Math. Soc. 64, 205-233.
CHUNG, K. L. - ERD6S, P.
A952) On the application of the Borel - Cantelli lemma. Trans. Am. Math. Soc.
72, 179-186.
CHUNG, K. L. - HUNT, G. A.
A949) On the zeros of ?"±1. Annals of Math. 50, 385-400.
CSAKI, E.
A978) On the lower limits of maxima and minima of Wiener process and partialsums. Z. Wahrscheinlichkeitstheorie verw. Gebiete 43, 205-221.
A980) A relation between Chung's and Strassen's law of the iterated logarithm.Z. Wahrscheinlichkeitstheorie verw. Gebiete 54, 287-301.
A989) An integral test for the supremum of Wiener local time. Probab. Th. Rel.
Fields 83, 207-217.
NON-ACTIVATEDVERSIONwww.avs4you.com
REFERENCES 315
CSAKI, E. - CSORGO, M. - FOLDES, A. - REVESZ, P.
A983) How big are the increments of the local time of a Wiener process? The
Annals of Probability 11, 593-608.
A989) Brownian local time approximated by a Wiener-sheet. The Annals ofProbability 17, 516-537.
CSAKI, E. - ERDOS, P. - REVESZ, P.
A985) On the length of the longest excursion. Z. Wahrscheinlichkeitstheorie
verw. Gebiete 68, 365-382.
CSAKI, E. - FOLDES, A.
A984/A) On the narrowest tube of a Wiener process. Coll. Math. Soc. J.
Bolyai 36, 173-197. Limit Theorems in Probability and Statistics (ed. P. Revesz)North-Holland.
A984/B) The narrowest tube of a recurrent random walk. Z. Wahrscheinlich-
Wahrscheinlichkeitstheorie verw. Gebiete 66, 387-405.
A984/C) How big are the increments of the local time of a simple symmetricrandom walk? Coll. Math. Soc. J. Bolyai 36, 199-221. Limit Theorems in
Probability and Statistics (ed. P. Revesz) North-Holland.
A986) How small are the increments of the local time of a Wiener process? The
Annals of Probability 14, 533-546.
A987) A note on the stability of the local time of a Wiener process. Stochastic
Processes and their Applications 25, 203-213.
A988/A) On the length of the longest flat interval. Proc. of the 5th Pannonian
Symp. on Math. Stat. 23-33 (ed. Grossmann, W. - Mogyorodi, J. - Vincze, I. -
Wertz, W.).
A988/B) On the local time process standardised by the local time at zero. Ada
Mathematica Hungartea. To appear.
CSAKI, E. - FOLDES, A. - KOMLOS, J.
A987) Limit theorems for Erdos - Renyi type problems. Studia Sci. Math. Hung.22, 321-332.
CSAKI, E. - FOLDES, A. - REVESZ, P.
A987) On the maximum of a Wiener process and its location. Probab. Th. Rel.
Fields 76, 477-497.
CSAKI, E. - GRILL, K.
A988) On the large values of the Wiener process. Stochastic Processes and their
Applications 27, 43-56.
NON-ACTIVATEDVERSIONwww.avs4you.com
316 REFERENCES
CSAKI, E. - REVESZ, P.
A979) How big must be the increments of a Wiener process? Ada Math. Acad.
Sci. Hung. 33, 37-49.
A983) A combinatorial proof of P. Levy on the local time. Ada Sci. Math.
Szeged 45, 119-129.
CSAKI, E. - VINCZE, I.
A961) On some problems connected with the Galton test. Publ. Math. Inst.
Hung. Acad. Sci. 6, 97-109.
CSORG6, M. - HORVATH, L. - REVESZ, P.
A987) Stability and instability of local time of random walk in random environ-
environment. Stochastic Processes and their Applications 25, 185-202.
CSORGO, M. - REVESZ, P.
A979/A) How big are the increments of a Wiener process? The Annals of Prob-
Probability 7, 731-737.
A979/B) How small are the increments of a Wiener process? Stochastic Processes
and their Applications. 8, 119-129.
A981) Strong Approximations in Probability and Statistics. Akademiai Kiado,Budapest and Academic Press, New York.
A985/A) On the stability of the local time of a symmetric random walk. Ada
Sci. Math. 48, 85-96.
A985/B) On strong invariance for local time of partial sums. Stochastic Processes
and their Applications 20, 59-84.
A986) Mesure du voisinage and occupation density. Probab. Th. Rel. Fields
73, 211-226.
DARLING, D. A. - ERDOS, P.
A956) A limit theorem for the maximum of normalized sums of independentrandom variables. Duke Math. J. 23, 143-145.
DAVIS, B.
A989) Loss of recurrence in reinforced random walk. Technical Report, Purdue
University.
A990) Reinforced random walk. Probab. Th. Rel. Fields. 84, 203-229.
DE ACOSTA, A.
A983) Small deviations in the functional central limit theorem with applicationsto functional laws of the iterated logarithm. The Annals of Probability 11, 78-
101.
NON-ACTIVATEDVERSIONwww.avs4you.com
REFERENCES 317
DEHEUVELS, P.
A985) On the Erdos - Renyi Theorem for Random Fields and Sequences and its
Relationship with the Theory of Runs and Spacings. Z. Wahrscheinlichkeitsthe-
orie verm. Gebiete 70, 91-115.
DEHEUVELS, P - DEVROYE, L. - LYNCH, I.
A986) Exact convergence rates in the limit theorem of Erdos - Renyi and Shepp.The Annals of Probability 14, 209-223.
DEHEUVELS, P. - ERD6S, P. - GRILL, K. - REVESZ, P.
A987) Many heads in a short block. Mathematical Statistics and ProbabilityTheory, Vol. A., Proc. of the 6th Pannonian Symp. 53-67 (ed. Puri, M. L. -
Revesz, P. - Wertz, W.).
DEHEUVELS, P. - REVESZ, P.
A986) Simple random walk on the line in random environment. Probability Th.
Rel. Fields 72, 215-230.
A987) Weak laws for the increments of Wiener processes, Brownian bridges,empirical processes and partial sums of i.i.d.r.v.'s. Mathematical Statistics and
Probability Theory, Vol. A., Proc. of the 6th Pannonian Symp. 69-88 (ed. Puri,M. L. - Revesz, P. - Wertz, W.).
DEHEUVELS, P. - STEINEBACH, J.
A987) Exact convergence rates in strong approximation laws for large increments
of partial sums. Probab. Th. Rel. Fields 76, 369-393.
DOBRUSHIN, R. L.
A955) Two limit theorems for the simplest random walk on a line. Uspehi Math.
Nauk (N. 5) 10, 139-146. In Russian.
DONSKER, M. D. - VARADHAN, S. R. S.
A977) On Laws of the Iterated Logarithm for Local Times. Comm. Pure Appl.Math. 30, 707-753.
A979) On the number of distinct sites visited by a random walk. Comm. Pure
Appl. Math. 27, 721-747.
DVORETZKY, A. - ERDOS, P.
A950) Some problems on random walk in space. Proc. Second Berkeley Sympo-Symposium 353-368.
DVORETZKY, A. - ERDOS, P. - KAKUTANI, S.
A950) Double points of Brownian paths in n-space. Ada Sci. Math. Szeged 12,75-81.
NON-ACTIVATEDVERSIONwww.avs4you.com
318 REFERENCES
ERD6S, P.
A942) On the law of the iterated logarithm. Annals of Math. 43, 419-436.
ERD6S, P. - CHEN, R. W.
A988) Random walks on Z%. J. Multivariate Analysis 25, 111-118.
ERD6S, P. - RENYI, A.
A970) On a new law of large numbers. J. Analyse Math. 23, 103-111.
ERD6S, P. - REVESZ, P.
A976) On the length of the longest head-run. Topics in Information Theory.Coll. Math. Soc. J. Bolyai 16, 219-228 (ed. Csiszar, I - Elias, P.).
A984) On the favourite points of a random walk. Mathematical Structures -
Computational Mathematics - Mathematical Modelling 2. Sofia, 152-157.
A987) Problems and results on random walks. Math. Statistics and ProbabilityTheory, Vol. B., Proc. 6th Pannonian Symp. 59-65 (ed. Bauer, P. - Konecny,F. - Wertz, W.) D. Reidel, Dordrecht.
A988) On the area of the circles covered by a random walk. Journal of Multi-
Multivariate Analysis 27, 169-180.
A989) A new law of the iterated logarithm. Ada Math. Hung. To appear.
ERDOS, P. - TAYLOR, S. J.
A960/A) Some problems concerning the structure of random walk paths. Ada
Math. Acad. Sci. Hung. 11, 137-162.
A960/B) Some intersection properties of random walk paths. Ada Math. Acad.
Sci. Hung. 11, 231-248.
FELLER, W.
A966) An Introduction to Probability Theory and Its Applications, Vol. II. J.
Wiley, New York.
A943) The general form of the so-called law of the iterated logarithm. Trans.
Am. Math. Soc. 54, 373-402.
FISHER, A.
A987) Convex - invariant means and a pathwise central limit theorem. Advances
in Mathematics 63, 213-246.
NON-ACTIVATEDVERSIONwww.avs4you.com
REFERENCES 319
FOLDES, A.
A975) On the limit distribution of the longest head-run. Matematikai Lapok 26,105-116. In Hungarian.
FOLDES, A. - PURI, M. L.
A989) The time spent by the Wiener process in a narrow tube before leaving a
wide tube. Preprint.
GNEDENKO, B. V. - KOLMOGOROV, A. N.
A954) Limit Distributions for Sums of Independent Random Variables. Addison- Wesley, Reading, Massachusetts.
GOLOSOV, A. O.
A983) Limit distributions for random walks in random environments. Soviet
Math. Dokl. 28, 18-22.
A984) Localization of random walks in one-dimensional random environments.
Commun. Math. Phys. 92, 491-506.
GONCHAROV, V. L.
A944) From the domain of Combinatorics. Izv. Akad. Nauk SSSR Ser. Math.
8A), 3-48.
GOODMAN, V. - KUELBS, J.
A988) Rates of convergence for increments of Brownian motion. J. of Theoretical
Probab. 1, 27-63.
GRIFFIN, P.
A989) Accelerating beyond the third dimension: Returning to the origin in sim-
simple random walk. The Mathematical Scientist. To appear.
GRILL, K.
A987/A) On the rate of convergence in Strassen's law of the iterated logarithm.Probab. Th. Rel. Fields 74, 583-589.
A987/B) On the last zero of a Wiener process. Mathematical Statistics and
Probability Theory ,Vol. A., 99-104 (ed. Puri, M. L. - Revesz, P. - Wertz, W.)
D. Reidel, Dordrecht.
GUIBAS, L. J. - ODLYZKO, A. M.
A980) Long repetitive patterns in random sequences. Z. Wahrscheinlichkeitsthe-
orie verw. Gebiete 53, 241-262.
HANSON, D. L. - RUSSO, R. P.
A983/A) Some results on increments of the Wiener process with applications to
lag sums of I.I.D. random variables. The Annals of Probability 11, 609-623.
NON-ACTIVATEDVERSIONwww.avs4you.com
320 REFERENCES
A983/B) Some more results on increments of the Wiener process. The Annals
of Probability 11, 1009-1015.
HARTMAN, P. - WINTNER, A.
A941) On the law of iterated logarithm. Amer. J. Math. 63, 169-176.
HAUSDORFF, F.
A913) Grundzuge der Mengenlehre. Leipzig.
HIRSCH, W. M.
A965) A strong law for the maximum cumulative sum of independent random
variables. Comm. Pure Appl. Math. 18, 109-217.
IMHOF, I. P.
A984) Density factorizations for Brownian motion meander and the three-
dimensional Bessel process. J. Appl. Probab. 21, 500-510.
ITO, K.
A942) Differential equations determining a MarkofF process. Kiyosi ltd Selected
Papers. Springer - Verlag, New York A986), 42-75.
ITO, K. - MCKEAN Jr., H. P.
A965) Diffusion processes and their sample paths. Die Grundlagen der Mathe-
matischen Wissenschaften Band 125. Springer - Verlag, Berlin.
KALIKOW, S. A.
A981) Generalized random walk in a random environment. The Annals of Prob-
Probability 9, 753-768.
KARLIN, S. - OST, F.
A988) Maximal length of common words among random letter sequences. The
Annals of Probability 16, 535-563.
KESTEN, H.
A965) An iterated logarithm law for the local time. Duke Math. J. 32, 447-456.
A980) The critical probability of band percolation on Z2 equals 1/2. Comm.
Math. Phys. 74, 41-59.
A987) Hitting probabilities of random walks on Zd. Stochastic Processes and
their Applications 25, 165-184.
A988) Recent progress in rigorous percolation theory. Asterisque 157-158, 217-
231.
NON-ACTIVATEDVERSIONwww.avs4you.com
REFERENCES 321
KESTEN, H. - SPITZER, F.
A979) A limit theorem related to a new class of self similar processes. Z.
Wahrscheinlichkeitstheorie verw. Gebiete 50, 5-25.
KEY, E. S.
A984) Recurrence and transience criteria for random walk in a random environ-
environment. The Annals of Probability 12, 529-560.
KHINCHINE, A.
A923) Uber dyadische Briiche. Math. Zeitachrift 18, 109-116.
KNIGHT, F. B.
A981) Essentials of Brownian Motion and Diffusion Am. Math. Soc, Provi-
Providence, R.I.
A986) On the duration of the longest excursion. Seminar on Stochastic Pro-
Processes, 1985. 117-147 Birkhauser, Boston.
KOLMOGOROV, A. N.
A933) Grundbegriffe der Wahrscheinlichkeitsrechnung. Springer, Berlin.
KOMLOS, J. - MAJOR, P. - TUSNADY, G.
A975) An approximation of partial sums of independent R.V.'s and the sampleDF. I. Z. Wahrscheinlichkeitstheorie verw. Gebiete 32, 111-131.
A976) An approximation of partial sums of independent R.V.'s and the sampleDF. II. Z. Wahrscheinlichkeitstheorie verw. Gebiete 34, 33-58.
LACEY, M. T. - PHILIPP, W.
A989) A note on the almost everywhere central limit theorem. Preprint.
LAMPERTI, J.
A977) Stochastic Processes. A Survey of the Mathematical Theory. Springer -
Verlag, New York.
LAWLER, G. F.
A980) A self-avoiding random walk. Duke Mathematical Journal 47, 655-692.
LE GALL, J.-F.
A988) Fluctuation results for the Wiener sausage. The Annals of Probability 16,991-1018.
NON-ACTIVATEDVERSIONwww.avs4you.com
322 REFERENCES
LEVY, P.
A948) Procesus Stochastique et Mouvement Brownien. Gauthier - Villars, Paris.
MAJOR, P.
A988) On the set visited once by a random walk. Probab. Th. Rel. Fields 77,117-128.
MCKEAN Jr, H. P.
A969) Stochastic Integrals. Academic Press, New York.
MOGUL'SKII, A. A.
A979) On the law of the iterated logarithm in Chung's form for functional spaces.
Th. of Probability and its Applications 24, 405-412.
MORI, T.
A989) More on the waiting time till each of some given patterns occurs as a run.
Preprint.
MUELLER, C.
A983) Strassen's Law for Local Time. Z. Wahrscheinlichkeitstheorie verw. Ge-
biete 63, 29-41.
NEMETZ, T - KUSOLITSCH, N.
A982) On the longest run of coincidences. Z. Wahrscheinlichkeitstheorie verw.
Gebiete 61, 59-73.
NEWMAN, D.
A984) In a random walk the number of "unique experiences" is two on the
average. SIAM Review 26, 573-574.
OREY, S. - PRUITT, W. E.
A973) Sample functions of the iV-parameter Wiener process. The Annals ofProbability 1, 138-163.
ORTEGA, I. - WSCHEBOR, M.
A984) On the increments of the Wiener process. Z. Wahrscheinlichkeitstheorie
verw. Gebiete 65, 329-339.
PERKINS, E.
A981) A global instrinsic characterization of Brownian local time. The Annals
of Probability 9, 800-817.
PETROV, V. V.
A965) On the probabilities of large deviations for sums of independent random
variables. Th. of Probability and its Applications 10, 287-298.
NON-ACTIVATEDVERSIONwww.avs4you.com
REFERENCES 323
PETROWSKY, I. G.
A935) Zur ersten Randwertaufgabe der Warmleitungsgleichung. Comp. Math.
B. 1, 383-419.
POLYA, G.
A921) liber eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt
im Strassennetz. Math. Ann. 84, 149-160.
QUALLS, G. - WATANABE, H.
A972) Asymptotic properties of Gaussian processes. Annals Math. Statistics 43,580-596.
RENYI, A.
A970/A) Foundations of Probability. Holden - Day, San Francisco.
A970/B) Probability Theory. Akademiai Kiado, Budapest and North Holland,Amsterdam.
REVESZ, P.
A978) Strong theorems on coin tossing. Proc. Int. Cong, of Mathematicians,Helsinki.
A979) A generalization of Strassen's functional law of iterated logarithm. Z.
Wahrscheinlichkeitstheorie verw. Gebiete 50, 257-264.
A982) On the increments of Wiener and related process. The Annals of Proba-
Probability 10, 613-622.
A988) In random environment the local time can be very big. Societe
Mathematique de France, Asterisque 157-158, 321-339.
A989/A) Simple symmetric random walk in Zd. Almost Everywhere Conver-
Convergence. Proceedings of the Int. Conf. on Almost Everywhere Convergence 369-392
(ed. G. A. Edgar, L. Sucheston) Academic Press, Boston.
A989/B) Estimates of the largest circle covered by a random walk. Preprint.
A989/C) On the volume of the balls covered by a random walk. Preprint.
RIESZ, F. - SZ. NAGY, B.
A953) Functional Analysis. Frederick Ungar, New York.
SAMAROVA, S. S.
A981) On the length of the longest head-run for a Markov chain with two states.
Th. of Probability and its Applications 26, 4B9-509.
NON-ACTIVATEDVERSIONwww.avs4you.com
324 REFERENCES
SCHATTE, P.
A988) On strong versions of the central limit theorem. Math. Nachr. 137,249-256.
SIMONS, G.
A983) A discrete analogue and elementary derivation of "Levy's equivalence" for
Brownian motion. Statistics & Probability Letters 1, 203-206.
SINAI, JA. G.
A982) Limit behaviour of one-dimensional random walks in random environment.
Th. of Probability and its Applications 27, 247-258.
SKOROHOD, A. V.
A961) Studies in the Theory of Random Processes. Addison - Wesley, Reading,Mass.
SOLOMON, F.
A975) Random walks in random environment. The Annals of Probability 3, 1-31.
SPITZER, F.
A958) Some theorems concerning 2-dimensional Brownian motion. Transactions
of the Am. Math. Soc. 87, 187-197.
A964) Principles of Random Walk. Van Nostrand, Princeton, N.J.
STRASSEN, V.
A964) An invariance principle for the law of iterated logarithm. Z. Wahrschein-
lichkeitstheorie verw. Gebiete 3, 211-226.
A966) A converse to the law of the iterated logarithm. Z. Wahrscheinlichkeits-
theorie verw. Gebiete 4, 265-268.
SZABADOS, T.
A989) A discrete Ito formula. Preprint.
SZEKELY, G. - TUSNADY, G.
A979) Generalized Fibonacci numbers, and the number of "pure heads". Matem-
atikai Lapok 27, 147-151. In Hungarian.
TOTH, B.
A985) A lower bound for the critical probability of the square lattice site perco-percolation. Z. Wahrscheinlichkeitstheorie verw. Gebiete 69, 19-22.
NON-ACTIVATEDVERSIONwww.avs4you.com
REFERENCES 325
TROTTER, H. F.
A958) A property of Brownian motion paths. Illinois J. of Math. 2, 425-433.
WEIGL, A.
A989) Zwei Sdtze uber die Belegungszeit beim Random Walk. Diplomarbeit, TU
Wien.
WICHURA, M.
A977) Unpublished manuscript.
ZIMMERMANN, G.
A972) Some sample function properties of the two-parameter Gaussian process.
Ann. Math. Statistics 43, 1235-1246.
NON-ACTIVATEDVERSIONwww.avs4you.com
Author Index
Andjel, E. D. 308, 309
Auer, P. 217, 222, 233, 235, 259
Bartfai, P. 53, 74
Bass, R. 130, 131
Berkes, I. 32
Bernoulli, J. xiii
Bickel, P. J. 170
Billingsley, P. 16
Bingham, N. H. 38
Bolthausen, E. 88, 311
Book, S. A. 67
Borel, E. 29
Borodin, A. N. 103
Brosamler, G. A. 146, 147
Chen, R. W. 59
Chung, K. L. 108, 114, 116, 142
Coppersmith, N. 312
Csaki, E. v, 18, 41, 44, 68, 70, 77, 78,84, 89, 90, 103, 108, 109, 110,111, 114, 115, 116, 117, 118,119, 120, 122, 123, 136, 137,139, 143, 144, 145, 157, 162,163, 164, 166, 169
Csorgo, M. v, 63, 69, 115, 116, 122,133, 134, 136, 137, 147, 175,279, 280
Darling, D. A. 170
Davis, B. 312
De Acosta, A. 89
Deheuvels, P. 18, 61, 71, 72, 75, 76,287, 288
Devroye, L. 76
Diaconis, P. 312
Dobrushin, R. L. 124, 125
Donsker, M. D. 209
Dvoretzky, A. 187, 195, 197, 207, 215
Erdos, P. v, 17, 18, 35, 39, 53, 57, 59,
62, 64, 74, 75, 116, 129, 130,142, 143, 144, 145, 170, 184,187, 195, 197, 200, 202, 205,206, 207, 213, 214, 215, 217,242
Feller, W. 19, 20, 35
Fisher, A. 146
Foldes, A. v, 18, 21, 70, 77, 78, 103,
114, 115, 116, 117, 118, 119,122, 123, 136, 137, 139, 145,160, 162, 163, 164
Gnedenko, B. V. 19
Golosov, A. O. 305, 306, 307
Goncharov, V. L. 21
Goodman, V. 90
Griffin, P. 130, 131, 186
Grill, K. 18, 44, 65, 66, 68, 75, 88,166, 168, 169
Guibas, L. J. 57, 59
Hanson, D. L. 69, 171
Horvath, L. 279, 280
Hunt, G. A. 108, 114
Imhof, I. P. 165
327
NON-ACTIVATEDVERSIONwww.avs4you.com
328 AUTHOR INDEX
Ito, K. 147, 173
Kakutani, S. 215
Kalikow, S. A. 309
Karlin, S. 63
Kesten, H. 108, 114, 259, 260, 306,
310, 311
Key, E. S. 307, 308
Khinchine, A. 30
Knight, F. B. 107, 145, 190, 191, 242
Kolmogorov, A. N. 19, 25, 35, 38
Komlos, J. 18, 53, 54
Kuelbs, J. 90
Kusolitsch, N. 63
Lacey, M. T. 147
Lamperti, J. 256
Lawler, G. F. 214
Le Gall, J. F. 211
Levy, P. 33, 100, 107, 146, 147
Lynch, I. 76
Major, P. 53, 54, 132
McKean Jr, H. P. 147, 175
Mogul'skii, A. A. 112
Mori, T. 59, 60
Mueller, C. 91, 121
Nemetz, T. 63
Newman, D. 132
Odlyzko, A. M. 57, 59
Orey, S. 194
Ortega, I. 64, 68
Ost, F. 63
Pascal, B. xiii
Perkins, E. 147
Petrov, V. V. 63
Petrowsky, I. G. 35
Philipp, W. 147
Polya, G. 23, 183
Pruitt, W. E. 194
Puri, M. L. 160
Quails, G. 170, 173
Renyi, A. 14, 19, 20, 28, 64, 74, 97
Revesz, P. 17, 18, 39, 57, 60, 62, 63,67, 68, 69, 71, 72, 75, 84, 87,105, 108, 109, 110, 111, 115,116, 120, 122, 129, 130, 133,134, 136, 137, 143, 144, 147,162, 163, 164, 175, 217, 218,227, 233, 235, 238, 241, 250,279, 280, 287, 288, 297
Riesz, F. 81
Rosenblatt, M. 170
Russo, R. P. 69, 171
Samarova, S. S. 57
Schatte, P. 146
Shore, T. R. 67
Simons, G. 109, 111
Sinai, JA, G. 276, 305, 306
Skorohod, A. V. 52
Solomon, F. 273
Spitzer, F. 27, 191, 192, 219, 242,310, 311
Steinebach, J. 76
Strassen, V. 32, 86
Szabados, T. 173
Szekely, G. 18
Sz. - Nagy, B. 81
Taylor, S. J. 145, 184, 195, 197, 200,
202, 205, 206, 213, 214, 215,242
Toth, B. 260
Trotter, H. F. 101
Tusnady, G. 18, 53, 54
Van Zwet, W. R. v
Varadhan, S. R. 209
NON-ACTIVATEDVERSIONwww.avs4you.com
AUTHOR INDEX 329
Varga, T. 55, 58
Vincze, I. 109
Watanabe, H. 170, 173
Weigl, A. 147
Wichura, M. 91
Wschebor, M. 64, 68
Zimmermann, G. 133
NON-ACTIVATEDVERSIONwww.avs4you.com
Subject Index
Arcsine law 100
Asymptotically deterministic sequence
34
Bernstein inequality 13
Borel - Cantelli lemma 27
Brownian motion 9
Chebyshev inequality 28
Dirichlet problem 255
DLA model 258
EFKP LIL 35
Gap method 29
Invariance principle 52, 105, 189
Ito formula 173
Ito integral 173
Large deviation theorem 19
Levy classes 33
LIL of Hartman and Wintner 32
LIL of Khinchine 31
Logarithmic density 146, 205
Markov inequality 28
Method of high moments 29
Normal numbers 29
Ornstein - Uhlenbeck process 169
Percolation 259
Quasi asymptotically deterministic se-
sequence 34
Rademacher functions 9, 11
Random walk in random environment
definition 265
local time 278, 292, 297
maximum 275, 277, 287, 305
recurrence 273
Random walk in random scenery 310
Random walk in Zx
definition 9
excursion 143, 147, 152
favourite values 129
first recurrence 94
increments 17, 73
increments of the local time 118
law of the iterated logarithm 31
law of the large numbers 28
local time 95, 100, 105, 113, 122,152
location of the last zero 98, 100,
141, 165
location of the maximum 98,100,142, 161
longest run 21, 55
longest zero-free interval 141
maximum 14, 20, 31, 35, 41
maximum of the absolute value
17, 20, 31, 35, 41, 163
mesure du voisinage 148
number of crossings 96, 109
range 44
rarely visited points 132
recurrence 23
Strassen type theorems 80, 86, 88,120
Random walk in Zd
completely covered balls 241
completely covered discs 217
definition 181
331
NON-ACTIVATEDVERSIONwww.avs4you.com
332 SUBJECT INDEX
favourite points 232, 247 law of the iterated logarithm 193
first recurrence 197 maximum 192
law of the iterated logarithm 193 rate of escape 195
local time 197, 231 selfcrossing 213
maximum 192 Strassen type theorems 194
range 207 Wiener sausage 210
rate of escape 195 Wiener sheet 133
recurrence 183 Zero-one law 25
selfcrossing 213
. speed of escape 249
Strassen type theorems 194
Reflection principle 15
Reinforced random walk 311
Skorohod embedding scheme 52
Tanaka formula 175
Theorem of Borel 28
Theorem of Chung 39
Theorem of Donsker and Varadhan
121
Theorem of Hausdorff 30
Theorem of Hirsch 39
Wichura's theorem 90
Wiener process in Rldefinition 48
excursion 145, 147
increments 63
increments of the local time 115
local time 100, 105, 187
location of the last zero 169
location of the maximum 165
longest zero-free interval 116
maximum 53
maximum of the absolute value
53
mesure du voisinage 147
occupation time 100
Strassen type theorems 80, 86, 88,120
Wiener process in Rddefinition 189
NON-ACTIVATEDVERSIONwww.avs4you.com