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

“Four legs
good,
two legs better
”
A
modified version
of
the
Animal Farm’s Constitution.
“Two logs
good,
p
logs
better
”
The original Constitution
of
mathematicians.
RAN D
OM
WALK
IN
RANDOM
AND
NO
N-
RAND
OM
ENVIRONMENTS
hECDND

EDITION
This page intentionally left blank
L
N
N-
M
ENVIRO
ECOND
0
Pal
Revesz
Technische Universitat Wien, Austria
Technical University
of
Budapest, Hungary
0
World
Scientific
1;
NEW JERSEY
*
LONDON
*
SINGAPORE
-
BEIJING
*
SHANGHAI
-
HONG

KONG
*
TAIPEI
-
CHENNAI
RANDEOM WALK
IN RANDOM AND
NON - RANDOM
ENVIRONMENTS
EDITION
Published by
World Scientific Publishing Co.
Pte.
Ltd.
5
Toh Tuck Link, Singapore 596224
USA
ofice;
27 Warren Street, Suite 401-402, Hackensack,
NJ
07601
UK
ofice:
57
Shelton Street, Covent Garden, London WC2H 9HE
Library
of
Congress Cataloging-in-Publication Data
Random walk in random and non-random environments
/

Pfll RCvCsz 2nd ed.
p. cm.
Includes bibliographical references and indexes.
ISBN
981-256-361-X (alk. paper)
1.
Random walks (Mathematics).
I.
Title.
QA274.73 .R48 2005
5
19.2’82 dc22
2005045536
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Copyright
0
2005 by World Scientific Publishing
Co.
Pte. Ltd.
All
rights reserved. This book, or parts thereof; may not be reproduced in any form or by any means,
electronic or mechanical, including photocopying, recording or any information storage and retrieval
system now known or to be invented, without written permission from the Publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright
Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA.
In
this case permission to
photocopy is
not

required from the publisher.
Printed in Singapore by Mainland Press
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 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-
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 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 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 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. Rkvksz
Technical University of Vienna
Wiedner Hauptstrasse 8-10/107
-4-1040 Vienna
Austria
V
This page intentionally left blank
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 to appear
in your subject and your book is going to be outdated. If you write
a
monograph on
a
very well-developed subject in which nothing new happens,
then it is going to 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 to 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 to
E.
CsAki,
M.
Csorgo”,
A.
Foldes,
D.
Khoshnevisan,
Y.
Peres,
Q.
M.
Shao,
B.
T6th,
Z.
Shi.
Vienna,
2005.
vii

This page intentionally left blank
Contents
Preface to the First Edition
V
Preface
to
the Second Edition
vii
Introduction xv
I
.
SIMPLE SYMMETRIC RANDOM
WALK
IN
Z’
Notations and abbreviations
3
1
Introduction
of
Part
I
9
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
CONTENTS
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

Zd
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
Z2

20.2
[(n)
in
Zd

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
23 Crossing and Self-crossing 241
24 Large Covered Balls
245
245

263
264
265
277
24.1
Completely covered discs centered in the origin
of
Z2

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
Zd

272
24.6
Large empty balls

24.7
Summary
of
Chapter
24

280
25 Long Excursions 281
25.1
Long excursions in
Z2

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)


CONTENTS
Xlll
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
Zd
.
. . .
.
.
.
. .
. .
.
.
.
. .
.
.
. .
.
. .
.

.
.
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
Subject Index 379
This page intentionally left blank
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
Zd)
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 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
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 to 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
This page intentionally left blank
Notations and abbreviations
Not
ations
General notations
1.
2.
3
4.
5.
6.
7.
8.
XI,
Xa, .
.
.
is a sequence of independent, identically distributed ran-
dom variables with
P{XI

=
l}
=
P{XI
=
-1}
=
1/2.
so
=
o,sn
=
S(n)
=
x1
-f-
x,
+
. .
'
+
xn
(n
=
1,2,.
.
.).
{Sn}
is the (simple symmetric) random walk.
ML

=
M+(n)
=
max
Sk,
Mi
=
M-(n)
=
-
min
Sk,
Mn
=
M(n)
=
max
lskl
=
max(Mz,M;),
M:
=
M:
+
Ml,
Yn
=
Mz
-
S,.

{W(t);t
2
0)
is
a
Wiener process (cf. Section
6.3).
O<k<n
O<k<n
O<k<n
m+(t)
=
sup
W(s),
o<s<t
m-(t)
=
-
inf
W(s),
m(t)
=
sup
JW(s)l
=
max(m+(t),m-(t))
(t
2
o),
ossit

ojs<t
m*(t)
=
m+(t)
+
m-(t),
y(t)
=
m+(t)
-
W(t).
b,
=
b(n)
=
(2nloglogn)-1/2,
Tn
=
r(n,a)
=
(2a
(log;
n
+
loglogn
))
-ll2.
[x]
is the largest integer less than or equal to
x.

3
4
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
I.
SIMPLE SYMMETRIC RANDOM
WALK
IN
Z’
Sometimes we use the notation
f(n)
-
g(n)
without any exact math-
emat,ical meaning, just saying that
f(n)
and
g(n)
are close to each
other in some sense.
1”

a(.)
=
-
1
e-U2/2du
is the standard normal distribution func-
-03
tion.
N
E
N(rn,a)
+j
P{o-l(N
-
rn)
<
x}
=
@(x).
#{.
.
.)
=
I{.
.
.}I
is the cardinality of the set in the bracket.
Rd
resp.
Zd

is the d-dimensional Euclidean space resp. its integer
grid.
B
=
Bd
is the set
of
Borel-measurable sets of
Rd.
A(.)
is
the Lebesgue measure on
EXd.
log,
(p
=
1,2,.
.
.)
is
pth iterated of log and lg resp.
logarithm resp.
pth
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
distributions
of
{
Vn}.
lg,
is
the
2,
Notations to the increments

NOTATIONS AND ABBREVIATIONS
5
6.
Jl(t,
U)
=
SUP
(W(S
+
U)
-
W(S)),
Ossst-a
7.
J2(t,u)
=
sup
IW(s
+
a)
-
W(s)l,
O<s<t-a
8.
J3(t,~)
=
SUP
SUP
(W(S
+

U)
-
W(S)),
O<s<t-a Osu<a
9.
J4(t,u)
=
sup sup
IW(s
+u)
-
W(s)l,
O<s<t-a O<u<a
11.
2,
is
the largest integer for which
I1(n,Zn)
=
Z,,
i.e.
2,
is the
length
of
the longest run of pure heads in
n
Bernoulli trials.
Notations to the Strassen-type theorems
2.

Wt(X)
=
btW(tz)
(0
5
II:
5
1,
t
>
O),
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
1.
c$(z,n)
=
#{k
:
0
<
k
5
72,
sk
=
X}
(X
=
0,
fl,
f2,.
.
.
,
n
=
1,2,.
.
.)

is the local time of the random walk
{sk}.
For any
A
C
Z1
we define
the occupation time
=(A,
n)
=
CzEA
<(x,
n).
9.3).
2.
V(X,
t)
(-co
<
x
<
+co,
t
2
0)
is the local time of
W(.)
(cf. Section
3.

H(A,
t)
=
X{s
:
0
5
s
5
t,
W(s)
E
A}
(A
c
R1
is
a
Bore1
set,
t
2
0)
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
>
p1,
Sk
=
0}
, ,
pn
=
min{k
:
k
>
pn-1,
sk
=o},
.

6
I.
SIMPLE SYMMETRIC RANDOM WALK IN
Z'
5.
For any
z
=
0,
*l,
k2,.
.

.
consider those values of
k
for which
Sk
=
z.
Let these values in increasing order be
0
<
PI(%)
<
p2(z)
<
. .
.
i.e.
pl(z)
=
min{k
:
k
>
0,Sk
=
z),p2(z)
=
min{k
:
k

>
pI(z),Sk
=
x},
. .
.
,p,(z)
=
min{k
:
k
>
p,-I(z),
Sk
=
z}
.
. .
Clearly
pi(0)
=
pi.
In case of a Wiener process define
p:
=
inf{t
:
t
2
O,q(O,t)

2
u).
6.
[(n)
=
max,
[(z,
n).
7.
dt)
=
SUP,
q(z1
t>.
8.
The random sequences
El
=
{SO,
S17
. .
Sp,
}I
E2
=
{spl]
Spl+l,
.
.
.,

SpZ}
1
. . .
are called the first, second,

excursions (away from
0)
of the random
walk
{Sk}.
9.
The random sequences
El(X)
=
{~p,(,),sp,(z)+l~'~
*
?SPZ(Z)h
E2(z)
=
{Sp2(z),
spz(Z)+ll~
'.
7
Sp3(z)I1
.
are called the first, second,

excursions away from
z
of the random

walk
{Sk}.
10. For any
t
>
0
let
a(t)
=
SU~{T
:
T
<
t,
W(T)
=
0)
and
P(t)
=
inf{.r
:
T
>
t,
W(T)
=
0).
Then the path
{Wt(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.
NOTATIONS AND ABBREVIATIONS
7
18.
+(t)
=
sup{s
:
s

>
0
for which there exists
a
0
<
u
<
t
-
s
such that
m+(u
+
s)
=
m+(.)}.
19.
R*(n)
=
max{k
:
k
>
1
for which there exists
a
0
<
j

<
n
-
k
such
that
M(j
+
Ic)
=
M(j)}.
20.
r*(t)
=
sup{s
:
s
>
0
for which there exists
a
0
<
u
<
t
-
s
such that
m(u

+
s)
=
m(u)}.
21.
p(n)
is the location of the maximum of the absolute value of
a
random
walk
{Sk}
up to
n,
i.e.
p(n)
is defined
by
S(p(n))
=
M(n)
and
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.
Mf(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))
=
MC(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.
This page intentionally left blank