RAN D OM WALK IN RANDOM AND NO N- RAND OM ENVIRONMENTS h E C D N D E D I T I O N pdf

15 A Few Further Results 15.1 On the location of the maximum of a random walk.. SIMPLE SYMMETRIC RANDOM WALK IN Z d Notations 17 The Recurrence Theorem 18 Wiener Process and Invariance

A modified version of the Animal Farm’s Constitution

“Two logs good, p logs better ”

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Preface to the First Edition

“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 (1981) where

it was told that the idea of that book was born in an inn in London over

a beer In spite of this danger Willem was brave enough t o invite me t o 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- ported by Fonds zur Forderung der Wissenschaftlichen Forschung, Project

Nr P6076 During these years I had very strong contact with the Math- ematical Institute of Budapest I am especially indebted t o 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 I1 are fully or partly due to him, but he had a significant influence even on those results that appeared under my name only

Last but not least, I have t o 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

Technical University of Vienna Wiedner Hauptstrasse 8-10/107

-4-1040 Vienna Austria

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Preface to the Second Edition

If you write a monograph on a new, just developing subject, then in the next few years quite a number of brand-new papers are going t o appear

in your subject and your book is going t o be outdated If you write a monograph on a very well-developed subject in which nothing new happens, then it is going t o be outdated already when it is going to appear In 1989 when I prepared the First Edition of this book it was not clear for me that its subject was already overdeveloped or it was a still developing area

A year later Erd6s told me that he had been surprised to see how many interesting, unsolved problems had appeared in the last few years about the very classical problem of coin-tossing (random walk on the line) In fact Erdos himself proposed and solved a number of such problems

I was happy to see the huge number of new papers (even books) that have appeared in the last 16 years in this subject I tried t o collect the most interesting ones and to fit them in this Second Edition Many of my friends helped me to find the most important new results and to discover some of the mistakes in the First Edition

My special thanks t o E CsAki, M Csorgo”, A Foldes, D Khoshnevisan,

Y Peres, Q M Shao, B T6th, Z Shi

Vienna, 2005

vii

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Contents

1.1 Randomwalk 9

1.2 Dyadic expansion 10

1.3 Rademacher functions 10

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 F’rom 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 the law of iterated logarithm 29

4.4 The LIL of Khinchine 31

Between the strong law of large numbers and 5 Lbvy Classes 33 5.1 Definitions 33

5.2 EFKPLIL 34

5.3 The laws of Chung and Hirsch 39

5.4 When will S, be very large? 39

ix

x C O N T E N T S

5.5 A theorem of Csaki

6 Wiener Process and Invariance Principle 6.1 Four lemmas

6.2 Joining of independent random walks

6.3 Definition of the Wiener process

6.4 Invariance Principle

7 Increments 7.1 Long head-runs

7.2 The increments of a Wiener process

7.3 The increments of 5 ’ ~

8 Strassen Type Theorems 8.1 The theorem of Strassen

8.2 Strassen theorems for increments

8.3 The rate of convergence in Strassen’s theorems

8.4 A theorem of Wichura

9 Distribution of the Local Time 9.1 Exact distributions

9.2 Limit distributions

9.3 Definition and distribution of the local time of a Wiener process

10 Local Time and Invariance Principle 10.1 An invariance principle

10.2 A theorem of LBvy

11 Strong Theorems of the Local Time 11.1 Strong theorems for [(z n) and [(n)

11.2 Increments of V(Z t )

11.3 Increments of <(z n)

11.4 Strassen type theorems

11.5 Stability

12 Excursions 12.1 On the distribution of the zeros of a random walk

12.2 Local time and the number of long excursions (Mesure du voisinage)

12.3 Local time and the number of high excursions

12.4 The local time of high excursions

12.5 How many times can a random walk reach its maximum?

41

47

47

49

51

52

57

57

66

77

83

90

92

95 a3

97

97

103

104

109

109

111

117

117

119

123

124

126

135

135

141

146

147

152

CONTENTS xi

13 F'requently and Rarely Visited Sites

13.1 Favourite sites

13.2 Rarely visited sites

14 An Embedding Theorem 14.1 On the Wiener sheet

14.2 The theorem

14.3 Applications

15 A Few Further Results 15.1 On the location of the maximum of a random walk

15.2 On the location of the last zero

15.3 The Ornstein-Uhlenbeck process and a theorem of Darling and Erd6s

15.4 A discrete version of the It6 formula

16 Summary of Part I I1 SIMPLE SYMMETRIC RANDOM WALK IN Z d Notations 17 The Recurrence Theorem 18 Wiener Process and Invariance Principle 19 The Law of Iterated Logarithm 20 Local Time 20.1 ( ( 0 n) in Z 2

20.2 [ ( n ) in Z d

20.3 A few further results

21 The Range 21.1 The strong law of large numbers

21.2 CLT LIL and Invariance Principle

2 1.3 Wiener sausage

22 Heavy Points and Heavy Balls 22.1 The number of heavy points

22.3 Heavy balls around heavy points

22.4 Wiener process

22.2 Heavy balls

157

157

161

163

163

164

168

171

171

175

179

183

187

191

193

203

207

211

211

218

220

221

221

225

226

227

227

236

239

240

xii CONTENTS

245

263

264 265 277 24.1 Completely covered discs centered in the origin of Z 2

24.2 Completely covered disc in iZ2 with arbitrary centre 24.3 Almost covered disc centred in the origin of Z2

24.4 Discs covered with positive density in 2’

24.5 Completely covered balls in Z d 272

24.6 Large empty balls

24.7 Summary of Chapter 24 280

25 Long Excursions 281 25.1 Long excursions in Z 2 281

25.2 Long excursions in high dimension 284

26 Speed of Escape 287 27 A Few Further Problems 293 27.1 On the Dirichlet problem 293

27.2 DLA model 296

27.3 Percolation 297

I11 RANDOM WALK IN RANDOM ENVIRONMENT Not at ions 301 28 Introduction 303 29 In the First Six Days 307 30 After the Sixth Day 311 30.1 The recurrence theorem of Solomon 311

30.2 Guess how far the particle is going away in an RE 313 30.3 A prediction of the Lord 314

30.4 A prediction of the physicist 326

31 What Can a Physicist Say About the Local Time <(O n)? 329 31.1 Two further lemmas on the environment 329

330 31.2 On the local time [ ( O n )

C O N T E N T S X l l l 32 On the Favourite Value of the RWIRE 33 A Few Further Problems 33.1 Two theorems of Golosov

33.2 Non-nearest-neighbour random walk

33.3 RWIRE in Z d

33.4 Non-independent environments

33.5 Random walk in random scenery

33.6 Random environment and random scenery

33.7 Reinforced random walk

References

Author Index

337

345

345

347

348

350

350

353

353

357

375

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Introduction

The first examinee is saying: Sir, I did not have time enough to study everything but 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 (or equivalently, coin tossing) the reader might say that this is a book for specialists written 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.) investi- gated the properties of coin tossing sequences and other simple games only Later on the progress of the probability theory went into two different di- rections:

(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 Nowadays the second direction is much more popular than the first one

I hope that:

(a) using the advantage of the simple situation coming from concen- trating 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 Z d ) is the simplest mathemati- cal model of the Brownian motion, the reader can find a simple way to the problems (at least to the classical problems) of statistical physics,

complicated sequences or processes

In spite of this fact this book mostly follows direction (i)

xv

xvi INTRODUCTION

(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 proper- ties 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 t o 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 strong theorems - 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 t o the location of the (or of a) proof

“The earth was without form and void, and dark- ness was upon the face of the deep.”

The First Book of Moses

I SIMPLE SYMMETRIC

RANDOM WALK IN Z1

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Notations and abbreviations

#{ .) = I{ .}I is the cardinality of the set in the bracket

Rd resp Z d is the d-dimensional Euclidean space resp its integer grid

B = Bd is the set of Borel-measurable sets of Rd

Ặ) is the Lebesgue measure on EXd

log, ( p = 1 , 2 , .) is p t h iterated of log and lg resp

logarithm resp p t h iterated of the logarithm of base 2

Let {U,} and {Vn} be two sequences of random variables

{U,, n = 1 , 2 , .} = {V, n = 1 , 2 , ) if the finite dimensional dis-

tributions of {U,} are equal to the corresponding finite dimensional

3 C(0,l) is the set of continuous functions defined on the interval [0,1],

4 S ( 0 , l ) is the Strassen’s class, containing those functions f(.) E C(0,l) for which f(0) = 0 and J;(~‘(x))’~x 5 1

Notations to the local time

is the occupation time of W ( ) (cf Section 9.3)

4 Consider those values of k for which Sk = 0 Let these values in increasing order be 0 = po < pl < p2 < , i.e p1 = min{k :

k > 0, 5’1, = 0}, p2 = min{k : k > p 1 , Sk = 0} , , pn = min{k :

k > pn-1, sk =o},

6 I SIMPLE S Y M M E T R I C R A N D O M W A L K IN Z'

5 For any z = 0, *l, k 2 , consider those values of k for which S k = z

Let these values in increasing order be 0 < PI(%) < p2(z) < i.e

p l ( z ) = min{k : k > 0,Sk = z ) , p 2 ( z ) = min{k : k > p I ( z ) , S k =

x}, ,p,(z) = min{k : k > p,-I(z), Sk = z} Clearly p i ( 0 ) = pi

In case of a Wiener process define p: = inf{t : t 2 O,q(O,t) 2 u)

10 For any t > 0 let a ( t ) = S U ~ { T : T < t , W ( T ) = 0) and P ( t ) = inf{.r :

T > t , W ( T ) = 0) Then the path { W t ( s ) ; a ( t ) 5 s 5 P ( t ) ) is called

an excursion of W ( )

11 c, is the number of those terms of S1, S2, , S, which are positive

or which are equal to 0 but the preceding term of which is positive

12 O ( n ) = # { k : 15 Ic 5 n, Sk-lSk+l < 0) is the number of crossings

13 R(n) = max{k : k > 1 for which there exists a 0 < j < n - k such

that c ( O , j + k ) = ( ( 0 , j ) ) is the length of the longest zero-free interval

14 r ( t ) = sup{s : s > 0 for which there exists a 0 < u < t - s such that 7d0,21+ s) = v(0, u)}

15 Q ( n ) = max{k : 0 5 k 5 n , Sk = 0) is the location of the last zero

up to n

16 $ ( t ) = sup{s : 0 < s 5 t , W ( s ) = 0)

17 R(n) = max{k : 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

p ( n ) 5 n If there are more integers satisfying the above conditions

then the smallest one will be considered as p ( n )

22 M ( t ) = inf{s : 0 < s 5 t for which W ( s ) = m ( t ) }

23 p + ( n ) = inf{k : 0 5 k 5 n for which S ( k ) = M + ( n ) }

24 M f ( t ) = inf{s : 0 < s 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, 4 , , S, is reached, i.e x(n) is the largest positive integer for which there exists a sequence of integers 0 5 kl < Icz < <

Icx(n) 5 n such that

S(lcl) = S(lC2) = = s(kx(n)) = M C ( 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.0 = infinitely often,

6 a.s = almost surely

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Chapter 1

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 toss- ing and his gain And a probabilist will consider independent, identically distributed 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

Consider a particle making a random walk (Brownian motion) on the real line Suppose that the particle starts from II: = 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 t o the right with equal probabilities 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 S, be the location of the particle after n steps or in time n This

model clearly implies that

P{S,+I = in+l I s, = in, sn-1 = i,-l, , s 1= i l , s o = io = 0)

where io = 0, i l , iz, , in, in+l is a sequence of integers with J i l - i 0 J =

J i 2 - i l l = = lin+l - i,l = 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

9

10 CHAPTER 1

first n steps It means that we consider

M n = max I s k ( resp M L = max s k or M L = - min s k

s - ' s - ' 0 - 20) = 2-(n+1)

where io = 0, i l , i 2 , , in+l is a sequence of integers with lil - iol =

liz - i l l = = lin+l - in1 = 1 Clearly (1.3) is equivalent t o (1.1) 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 N,(s) = Cy=l E ~ ( z ) of the ones among the first n digits of z E [0,1] Since N,(z) = ( n - Sn(z))/2 any theorem formulated for S, implies a corresponding theorem for N n ( z )

X{z: T j l ( Z ) = & , r&) =&, , Tj&) = s , > = 2 - , (1.4)

where 1 5 j l < j z < < j , ( n = 1 , 2 , .); & , & , ,6, is an arbitrary sequence of +l’s and -1’s and X is the Lebesgue measure Putting So =

So(z) = 0 and S, = S,(z) = Cy=l ri(z) ( n = 1 , 2 , .) we obtain (1.3)

Two gamblers (A and B) are tossing a coin A wins one dollar if the tossing results in a head and B wins one dollar if the result is tail Let S, be the amount gained by A (in dollars) after n tossings (Clearly S, can be negative and So = 0 by definition.) Then SN satisfies (1.1) if the game is fair, i.e the coin is regular

Let X 1 , X z , be a sequence of i.i.d.r.v.’s with

Then (1.5) implies that {S,} is a Markov chain, i.e

P{S,+1 = in+l 1 s, = in, S,-l = & - I , , Sl = i l , so = io = 0 )

where i,-, = 0, i l , i 2 , ,in,in+l is a sequence of integers with Jil - iol =

t i 2 - ill = = (in+l - in( = 1

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t E rw]- we have

The following inequality (Bernstein inequality) can also be obtained by elementary methods:

for any n = 1 , 2 , and 0 < E 5 1/4

14 CHAPTER 2

For later reference we present also a slightly more general form of the Bernstein inequality

quence of i.i.d.r.u with

P{X, = 1) = 1 - P{x; = 0) = p

T h e n f o r a n y 0 < E 5 pq we have

where S: = X; +X,* + +X: and q = 1 - p

A slightly more precise form of the above Theorem is the so-called

L A R G E D E V I A T I O N THEOREM (cf Durrett, 1991, p 61) Let

D I S T R I B U T I O N S 15 Similarly for k = 0

1

~ n + l , o = Pix1 = -1, ML 5 1) = Z(pn,l +pn,o) (2.8) Since p l , ~ = p1,l = 1/2 we get (2.5) from (2.7) and (2.8) by induction

and we obtain (2.10) easily Assume that (2.10) holds for n - 1 and for

any a , b, v satisfying the conditions of the Theorem Now we prove (2.10)

by induction Note that p,(O, b, v ) = p n ( a , 0, v ) = 0 and the same is true

for the righthand side of (2.10) (since the terms cancel because q n ( j ) =

q , ( - j ) ) Hence we may assume that a < 0 < b But in this case a + 1 5 0 and b - 1 _> 0 Hence by induction (2.10) holds with parameters n - 1, a +

1, b + 1, v and n - 1, a - 1, b - I, v We obtain (2.10) observing that

(2.11) is a simple consequence of (2.10), (2.12) follows from (2.11) taking

2~ = a , Y = b and (2.13) follows from (2.12) taking a = -b

To evaluate the distribution of I i ( 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 I1 ( n , a )

j + 2

p ( n + j , n ) := P{Il(n + j , n ) = n } = - 2n+l ( j = 0,172, , n )

Clearly p ( n + j , n ) is the probability that a coin tossing sequence of

length n + j contains a pure-head-run of length n

A = { I l ( n + j , n ) = n } and A k = {Sk+, - S k = n }

Then

A = A0 + Ao.41 + &A1 A2 + + AoAl A j _ l A j

= AO + AoAi + AiAz + * + Aj-IAj

Since P{Ao} = 2-, and P{AoAl AjAj+l} = 2-,-' for any j = 1 , 2 , ,

we have the Lemma

The next recursion can be obtained in a similar way

known, namely:

THEOREM 2.7 (Szkkely - Tusnady, 1979)

P{Zn < s} = 2-97>)

where

Remark 1 Csaki, Foldes and Koml6s (1987) worked out a very general

method to obtain inequalities like (2.14) Their method gives a somewhat weaker result than (2.14) However, their result is also strong enough to

produce most of the strong theorems given later (cf Section 7.3)

DISTRIBUTIONS 19

2.2 Limit distributions

Utilizing the Stirling formula

(where 0 < g n < 1) and the results of Section 2.1, the following limit theorems can be obtained

0 < E < 1 / 2 the inequality En < k < (1 - E)n is satisfied Then

where K = k/n and d ( K ) = Klog2K + (1 - K ) log2(1 - K ) If we also assume that I k - n/2 I= o(n2I3) then

Especially

The next theorem is the so-called Central Limit Theorem

THEOREM 2.9 (Gnedenko - Kolmogorov, 1954, 840)

sup lP{n-112Sn < 2) - +((.)I 5 2n-I”

X

A stronger version of Theorem 2.9 is another form of the Large Deviation Theorem:

provided that 0 < xn = o(n1/6)

Theorem 2.10 can be generalized as follows:

20 CHAPTER 2

THEOREM 2.11 (e.g Feller, 1966, p 517) Let X ; , X ; , be a se- quence of i.i.d.r.v 's with

EXz* = 0, E(XZ*)' = 1, E(exp(tX%*)) < co

for all t in some interval 1 t I< to Then

provided that 0 < x, = o(n1/6) where 5'; = X ; + X,* + + X:

THEOREM 2.12 (e.g Rknyi, 1970/A, p 234)

i f 0 < x n = o(n116) and n is large enough

Remark 1 As we claimed G ( x ) = H ( x ) however in Theorem 2.13 the

asymptotic distribution in the form of G(.) is proposed to be used when x

is large When z is small, H ( ) is more adequate

Finally we present the limit distribution of 2,

THEOREM 2.14 (Foldes, 1975, Goncharov, 1944) For any positive in-

(2.17)

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i.e t h e probability t h a t a particle starting from i hits 0 before k is k-' ( k - i )

Proof Clearly we have

p(O,O, k) = 1, p ( 0 , Ic, 5) = 0

When the particle is located in i then it hits 0 before k if

(i) either it goes to i - 1 (with probability 1/2) and from i - 1 goes to 0 before k (with probability p(0,i - 1, k)),

(ii) or it goes to i + 1 (with probability 1/2) and from i + 1 goes to 0 before k (with probability p ( 0 , i + 1, k ) )

1

2

That is

1

p(O,i, k) = zp(0,i - 1, k) + -p(O,i + 1, k)

( i = 1 , 2 , , k - 1) Hence p(O,i, k) is a linear function of i, being 1 in 0 and 0 in Ic, which implies (3.1)

23