Multiparameter Processes: An Introduction to Random Fields

605 Pages·2002·4.305 MB·English

by  Davar Khoshnevisan

S M M
pringer onographs in athematics

Davar Khoshnevisan
Multiparameter Processes
An Introduction to Random Fields
i
Springer
DavarKhoshnevisan
DepartmentofMathematics
UniversityofUtah
SaltLakeCity,Ut84112-0090
[email protected]
With12illustrations.
MathematicsSubjectClassification(2000):60Gxx,60G60
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Khoshnevisan,Davar.
Multiparameterprocesses:anintroductiontorandomfields/DavarKhoshnevisan.
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ISBN0-387-95459-7(alk.paper)
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Preface
Thisbookaimstoconstructageneralframeworkfortheanalysisofalarge
familyofrandomfields,alsoknownasmultiparameterprocesses.Theneed
for such a development was pointed out in Doob (1990, p. 47). Referring
to the theory of one-parameter stochastic processes, Doob writes:1
Ourdefinitionofastochasticprocessishistoricallyconditioned
andhasobviousdefects.Inthefirstplacethereisnomathemat-
ical reason for restricting T to be a set of real numbers, and in
fact interesting work has already been done in other cases. (Of
course,theinterpretationoftastimemustthenbedropped.)In
thesecondplacethereisnomathematicalreasonforrestricting
the value assumed by the xt’s to be numbers.
There are a number of compelling reasons for studying random fields,
one of which is that, if and when possible, multiparameter processes are a
natural extension of existing one-parameter processes. More exciting still
arethevariousinteractionsbetweenthetheoryofmultiparameterprocesses
andotherdisciplines,includingprobabilityitself.Forexample,inthisbook
thereaderwilllearnofvariousconnectionstorealandfunctionalanalysis,a
modicumofgrouptheory,andanalyticnumbertheory.Themultiparameter
processes of this book also arise in applied contexts such as mathematical
statistics (Pyke 1985), statistical mechanics (Kuroda and Manaka 1998),
and brain data imaging (Cao and Worsley 1999).
1Heisreferringtoastochasticprocessoftheform(xt; t∈T).
vi
My writing philosophy has been to strike a balance between developing
a reasonably general theory, while presenting applications and explicit cal-
culations. This approach should set up the stage for further analysis and
exploration of the subject, and make for a more lively approach.
Thisbookisintwoparts.PartIisaboutthediscrete-timetheory.Italso
contains results that allow for the transition from discrete-time processes
to continuous-time processes. In particular, it develops abstract random
variables, parts of the theory of Gaussian processes, and weak convergence
for continuous stochastic processes. Part II contains the general theory of
continuous-time processes. Special attention is paid to processes with con-
tinuous trajectories, but some discontinuous processes will also be studied.
In this part I will also discuss subjects such as potential theory for sev-
eral Markov processes, the Brownian sheet, and some Gaussian processes.
Parts I and II are influenced by the fundamental works of Doob, Cairoli,
and Walsh.
My goal has been to keep this book as self-contained as possible, in
order to make it available to advanced graduate students in probability
and analysis. To this I add that a more complete experience can only be
gainedthroughsolvingmanyoftheproblemsthatarescatteredthroughout
the body of the text. At times, these in-text exercises ask the student to
check some technical detail. At other times, the student is encouraged to
apply a recently introduced idea in a different context. More challenging
exercises are offered at the end of each chapter.
Many of the multiparameter results of this book do not seem to exist
elsewhere in a pedagogic manner. There are also a number of new theo-
remsthatappearhereforthefirsttime.Whenintroducingabetter-known
subject (e.g., martingales or Markov chains), I have strived to construct
the most informative proofs, rather than the shortest.
This book would not exist had it not been for the extensive remarks,
corrections, and support of R. Bass, J. Bertoin, K. Burdzy, R. Dalang, S.
Ethier,L.Horva´th,S.Krone,O.L´evˆeque,T.Lewis,G.Milton,E.Nualart,
T.Mountford,J.Pitman,Z.Shi,J.Walsh,andY.Xiao.Theireffortshave
led to a much cleaner product. What errors remain are my own. I have en-
joyedagreatdealoftechnicalsupportfromP.Bowman,N.Beebe,andthe
editorialstaffofSpringer.TheNationalScienceFoundationandtheNorth
Atlantic Treaty Organization have generously supported my work on ran-
domfieldsovertheyears.Mysincerestgratitudegoestothemall.Finally,I
wishtothankmydearestfriend,andmysourceofinspiration,IrinaGushin.
ThisbookisdedicatedtothememoryofVictorGushin,andtotherecent
arrival of Adrian V. Kh. Gushin.
Davar Khoshnevisan, Salt Lake City, UT.
March 2002
Contents
Preface v
List of Figures xv
General Notation xvii
I Discrete-Parameter Random Fields 1
1 Discrete-Parameter Martingales 3
1 One-Parameter Martingales . . . . . . . . . . . . . . . . . . 4
1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 The Optional Stopping Theorem . . . . . . . . . . . 7
1.3 A Weak (1,1) Inequality . . . . . . . . . . . . . . . . 8
1.4 A Strong (p,p) Inequality . . . . . . . . . . . . . . . 9
1.5 The Case p=1 . . . . . . . . . . . . . . . . . . . . . 9
1.6 Upcrossing Inequalities . . . . . . . . . . . . . . . . 10
1.7 The Martingale Convergence Theorem . . . . . . . . 12
2 Orthomartingales . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1 Definitions and Examples . . . . . . . . . . . . . . . 16
2.2 Embedded Submartingales . . . . . . . . . . . . . . 18
2.3 Cairoli’s Strong (p,p) Inequality . . . . . . . . . . . 19
2.4 Another Maximal Inequality . . . . . . . . . . . . . 20
2.5 A Weak Maximal Inequality . . . . . . . . . . . . . . 22
viii Contents
2.6 Orthohistories . . . . . . . . . . . . . . . . . . . . . 22
2.7 Convergence Notions . . . . . . . . . . . . . . . . . . 24
2.8 Topological Convergence . . . . . . . . . . . . . . . . 26
2.9 Reversed Orthomartingales . . . . . . . . . . . . . . 30
3 Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 Marginal Filtrations . . . . . . . . . . . . . . . . . . 31
3.3 A Counterexample . . . . . . . . . . . . . . . . . . . 33
3.4 Commutation . . . . . . . . . . . . . . . . . . . . . . 35
3.5 Martingales . . . . . . . . . . . . . . . . . . . . . . . 37
3.6 Conditional Independence . . . . . . . . . . . . . . . 38
4 Supplementary Exercises . . . . . . . . . . . . . . . . . . . . 40
5 Notes on Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . 44
2 Two Applications in Analysis 47
1 Haar Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 47
1.1 The 1-Dimensional Haar System . . . . . . . . . . . 48
1.2 The N-Dimensional Haar System . . . . . . . . . . . 51
2 Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.1 Lebesgue’s Differentiation Theorem. . . . . . . . . . 54
2.2 A Uniform Differentiation Theorem . . . . . . . . . 58
3 Supplementary Exercises . . . . . . . . . . . . . . . . . . . . 61
4 Notes on Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . 63
3 Random Walks 65
1 One-Parameter Random Walks . . . . . . . . . . . . . . . . 66
1.1 Transition Operators . . . . . . . . . . . . . . . . . . 66
1.2 The Strong Markov Property . . . . . . . . . . . . . 69
1.3 Recurrence . . . . . . . . . . . . . . . . . . . . . . . 70
1.4 Classification of Recurrence . . . . . . . . . . . . . . 72
1.5 Transience. . . . . . . . . . . . . . . . . . . . . . . . 74
1.6 Recurrence of Possible Points . . . . . . . . . . . . . 75
1.7 Recurrence–Transience Dichotomy . . . . . . . . . . 78
2 Intersection Probabilities. . . . . . . . . . . . . . . . . . . . 80
2.1 Intersections of Two Walks . . . . . . . . . . . . . . 80
2.2 An Estimate for Two Walks . . . . . . . . . . . . . . 85
2.3 Intersections of Several Walks . . . . . . . . . . . . . 86
2.4 An Estimate for N Walks . . . . . . . . . . . . . . . 89
3 The Simple Random Walk . . . . . . . . . . . . . . . . . . . 89
3.1 Recurrence . . . . . . . . . . . . . . . . . . . . . . . 90
3.2 Intersections of Two Simple Walks . . . . . . . . . . 91
3.3 Three Simple Walks . . . . . . . . . . . . . . . . . . 93
3.4 Several Simple Walks . . . . . . . . . . . . . . . . . 97
4 Supplementary Exercises . . . . . . . . . . . . . . . . . . . . 99
5 Notes on Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . 103
Contents ix
4 Multiparameter Walks 105
1 The Strong Law of Large Numbers . . . . . . . . . . . . . . 106
1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . 106
1.2 Commutation . . . . . . . . . . . . . . . . . . . . . . 107
1.3 A Reversed Orthomartingale . . . . . . . . . . . . . 109
1.4 Smythe’s Law of Large Numbers . . . . . . . . . . . 110
2 The Law of the Iterated Logarithm . . . . . . . . . . . . . . 112
2.1 The One-Parameter Gaussian Case . . . . . . . . . . 113
2.2 The General LIL . . . . . . . . . . . . . . . . . . . . 116
2.3 Summability . . . . . . . . . . . . . . . . . . . . . . 117
2.4 Dirichlet’s Divisor Lemma . . . . . . . . . . . . . . . 118
2.5 Truncation . . . . . . . . . . . . . . . . . . . . . . . 119
2.6 Bernstein’s Inequality . . . . . . . . . . . . . . . . . 121
2.7 Maximal Inequalities . . . . . . . . . . . . . . . . . . 123
2.8 A Number-Theoretic Estimate . . . . . . . . . . . . 125
2.9 Proof of the LIL: The Upper Bound . . . . . . . . . 127
2.10 A Moderate Deviations Estimate . . . . . . . . . . . 128
2.11 Proof of the LIL: The Lower Bound . . . . . . . . . 130
3 Supplementary Exercises . . . . . . . . . . . . . . . . . . . . 132
4 Notes on Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . 135
5 Gaussian Random Variables 137
1 The Basic Construction . . . . . . . . . . . . . . . . . . . . 137
1.1 Gaussian Random Vectors . . . . . . . . . . . . . . . 137
1.2 Gaussian Processes . . . . . . . . . . . . . . . . . . . 140
1.3 White Noise. . . . . . . . . . . . . . . . . . . . . . . 142
1.4 The Isonormal Process . . . . . . . . . . . . . . . . . 144
1.5 The Brownian Sheet . . . . . . . . . . . . . . . . . . 147
2 Regularity Theory . . . . . . . . . . . . . . . . . . . . . . . 148
2.1 Totally Bounded Pseudometric Spaces . . . . . . . . 149
2.2 Modifications and Separability . . . . . . . . . . . . 153
2.3 Kolmogorov’s Continuity Theorem . . . . . . . . . . 158
2.4 Chaining . . . . . . . . . . . . . . . . . . . . . . . . 160
2.5 H¨older-Continuous Modifications . . . . . . . . . . . 165
2.6 The Entropy Integral. . . . . . . . . . . . . . . . . . 167
2.7 Dudley’s Theorem . . . . . . . . . . . . . . . . . . . 170
3 The Standard Brownian Sheet. . . . . . . . . . . . . . . . . 172
3.1 Entropy Estimate . . . . . . . . . . . . . . . . . . . 172
3.2 Modulus of Continuity . . . . . . . . . . . . . . . . . 173
4 Supplementary Exercises . . . . . . . . . . . . . . . . . . . . 175
5 Notes on Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . 178
6 Limit Theorems 181
1 Random Variables . . . . . . . . . . . . . . . . . . . . . . . 181
1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . 182