Large Deviations

156 Pages·2008·12.769 MB·English

by  Frank den Hollander

http://dx.doi.org/10.1090/fim/014
FIELDS INSTITUTE
MONOGRAPHS
THE FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCE S
Large Deviation s
Frank den Hollander
American Mathematical Society
Providence, Rhode Island
The Fields Institut e
for Research in Mathematical Science s
The Fields Institute is a center for mathematical research, located in Toronto, Canada.
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Fields Institute Editorial Board: Car l R. Riehm (Managing Editor), Barbara Lee Key-
fitz (Directo r of the Institute), Juris Steprans (Deput y Director), John Blan d (Toronto) ,
Kenneth R . Davidso n (Waterloo) , Joe l Feldman (UBC) , R. Mar k Goresk y (Institut e fo r
Advanced Study , Princeton), Cameron Stewart (Waterloo) , Noriko Yui (Queen's) .
2000 Mathematics Subject Classification. Primar y 60-01 , 60F10, 60K35;
Secondary 82B31, 82B44.
ABSTRACT. This book is an introduction to large deviations and consists of two parts: Par t A
theory, Part B applications.
Part A describes the basic large deviation theorems for i.i.d. sequences, Markov sequences, and
sequences with moderate dependence. I t also gives an outline of definitions an d theorems in a
more abstract context, exposing the unified scheme that gives large deviation theory its overall
structure.
Part B describes a selection of applications, most of which are recent and circle around statistical
physics and random media.
The book contains some 60 exercises with solutions.
For additional information an d updates on this book, visi t
www.ams.org/bookpages/fim-14
Library of Congress Cataloging-in-Publication Dat a
Hollander, F. den (Prank )
Large deviations / Frank den Hollander.
p. cm. — (Fields Institute monographs, ISSN 1069-5273 ; 14)
Includes bibliographical references and index.
ISBN 0-8218-1989-5 (alk. paper)
1. Large deviations. I . Title. II . Series.
QA273.67.H65 200 0
519.5'34—dc21
99-058913
AMS softcover ISBN: 978-0-8218-4435-9
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© 200 0 by the American Mathematical Society. All rights reserved.
Reprinted by the American Mathematical Society, 2008.
The American Mathematical Society retains all rights
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Printed in the United States of America.
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10 9 8 7 6 5 4 3 2 1 7 16 15 14 13 12
Contents
PREFACE i x
Part A. THEOR Y 1
Chapter I. LARG E DEVIATIONS FOR I.I.D. SEQUENCES: PART 1 3
LI. Introductio n 3
1.2. A n example: coin tossing 4
1.3. Cramer' s Theorem for the empirical average 5
1.4. Comment s 8
1.5. A toy application of Cramer 1 0
Chapter II. LARG E DEVIATIONS FOR I.I.D. SEQUENCES: PART 2 1 3
ILL Sanov' s Theorem for the empirical measure 1 3
11.2. Th e pair empirical measure 1 5
11.3. A toy application of Sanov for pairs 1 8
11.4. A contraction principle 1 9
11.5. Th e empirical process 2 0
II. 6. Comment s 2 5
II. 7. Extensio n to countable state space 2 6
Chapter III. GENERA L THEORY 2 9
111.1. Th e large deviation principle (LDP) 2 9
111.2. Comment s 3 1
111.3. Varadhan' s Lemma 3 2
111.4. Th e LDP for integrals of exponential functionals 3 4
111.5. Th e Contraction Principle 3 5
111.6. Th e weak LDP 3 5
111.7. Convexit y 3 6
111.8. Relatio n to earlier results 4 1
Chapter IV. LARG E DEVIATIONS FOR MARKOV SEQUENCES 4 3
IV. 1. Radon-Nikody m formula 4 3
IV.2. Th e LDP for discrete-time Markov chains 4 4
IV. 3. Comment s 4 6
IV.4. Th e LDP for continuous-time Markov chains 4 7
Chapter V. LARG E DEVIATIONS FOR DEPENDENT SEQUENCES 5 3
V. 1. Preliminarie s 5 3
V.2. Th e Gartner-Ellis Theorem 5 4
V.3. Comment s 5 8
V.4. Relatio n to earlier results 6 0
vi Content s
V.5. Conclusio n 6 1
Part B. APPLICATION S 6 3
Chapter VI. STATISTICA L HYPOTHESIS TESTING 6 5
VI. 1. Th e statistical problem 6 5
VI.2. Larg e deviation estimates on test optimality 66
Chapter VII. RANDO M WALK IN RANDOM ENVIRONMENT 6 9
VII. 1. Rando m drifts 6 9
VII.2. Th e LDP for the speed 7 0
VII.3. Th e LDP for the hitting times 7 2
VII.4. Pro m hitting times to speed 7 3
VII.5. Rando m continued fractions 7 4
VII.6. Analysi s of the rate functions 7 5
VII.7. Compariso n with homogeneous random walk 7 7
VII.8. Concludin g remarks 7 9
Chapter VIII . HEA T CONDUCTION WIT H RANDO M SOURCE S AN D
SINKS 8 1
VIII. 1. Th e parabolic Anderson model 8 1
VIII.2. Growt h rate of the moments 8 3
VIII.3. Analysi s of the variational problem 8 4
VIII.4. Correlatio n structure 8 5
VIII.5. Derivatio n of the growth rate 8 7
VIII.6. Transformatio n of the variational problem 8 9
VIII.7. Concludin g remarks 9 1
Chapter IX. POLYME R CHAINS 9 3
IX. 1. A polymer model: self-repellent random walk 9 3
IX.2. Linea r speed 9 4
IX.3. Th e LDP for bridges 9 6
IX.4. Ste p 1: Adding drift 9 8
IX.5. Ste p 2: Markovian nature of the total local times 9 9
IX.6. Ste p 3: Key variational problem 10 0
IX.7. Ste p 4: Analysis of the rate function for bridges 10 2
IX.8. Ste p 5: Identification of the speed 10 4
IX.9. Th e LDP without the bridge condition 10 5
IX. 10. Concludin g remarks 10 8
Chapter X. INTERACTIN G DIFFUSIONS 11 1
X.l. Th e interaction Hamiltonian 11 1
X.2. Radon-Nikody m formula 11 2
X.3. Th e LDP for the double layer empirical measure 11 3
X.4. A simpler representation for the rate function 11 4
X.5. McKean-Vlaso v equation 11 5
X.6. Th e Kuramoto model 11 7
X.7. Concludin g remarks 12 0
Appendix: Solutions to the Exercises 12 1
Bibliography 135
Contents
Index 139
Glossary of Symbols 141
Errata 143
Errata for the Reprinted Edition 145
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PREFACE
Large deviation theory is a part of probability theory that deals with the de-
scription of events where a sum of random variables deviates from it s mean by
more than a "normal" amount, i.e., beyond what is described by the central limit
theorem. A precise calculation of the probabilities of such events turns out to be
crucial for the study of integrals of exponential functionals of sums of random vari-
ables, which come up in a variety of different contexts . Larg e deviation theor y
finds application in probability theory, statistics, operations research, ergodic the-
ory, information theory, statistical physics, financial mathematics, and the list goes
on.
These lecture notes are an introduction t o large deviations. Part A (Chapters I-
V) describes theory, Par t B (Chapters VI-X) describes applications. A glance at
the table of contents shows what topics are covered. I have put much effort int o
conveying the mai n idea s without puttin g to o muc h emphasis o n technicalities .
Most of the theory is driven by a few "key principles" and once these are understood
the rest of the journey is safe sailing, give or take a storm or two. Thi s is not t o
say that it is easy to grasp the full theoretical panorama. Bu t the reader's patience
will be rewarded when the ship enters the harbor of the applications.
Chapters I and II contain the basic large deviation theorems for i.i.d. random
variables. Her e the goal is to make the reader acquainted with the type of state-
ments that are typical for the theory and to obtain results via explicit calculation
of the rate function. Chapte r II I presents general definitions an d theorems in a
more abstract context . Her e the goal is to expose the unified schem e that give s
large deviation theory its overall structure and that can be made to work in many
concrete cases. Chapte r IV looks at large deviation theorems for Markov chain s
and explains how these can be obtained from the i.i.d. case via a change-of-measure
argument. Chapte r V considers random sequences with moderate dependence and
shows how many of the results in Chapters I, II and IV can be put under a single
heading, which in some sense closes the circle.
Chapters VI-X describe a selection of applications: statistical hypothesis test-
ing; random walk in random environment; hea t conductio n with random source s
and sinks; polymer chains ; interacting diffusions . Her e the theory comes to lif e
and the reader gets to see the full impact of the results derived earlier. Except for
the application in statistical hypothesis testing, which is put in mainly for didacti-
cal reasons, all applications are recent. Naturall y their choice reflects my personal
taste and involvement, since they circle around statistical physics and random me-
dia. But I think they offer a good sample of what large deviation theory is able to
do in various different contexts. Each application is self-contained and tells a small
story.
ix
x PREFAC E
Many questions that come up during the exposition are posed as exercises to
the reader. The solutions to these exercises are given in the Appendix. At the end
I have included a list of frequently used words and symbols, with the number of the
section where they appear first. Thi s will help the reader to connect the differen t
chapters.
Large deviation theory is a mixture of probability theory, convex analysis, vari-
ational calculus and set topology. A s such it is mathematically both challengin g
and captivating. Eve n so, it is not obvious how to do justice to a vast area like
large deviation theory in one hundred pages or so, especially when the goal is to
cover both theory and applications. T o focus ideas, I have restricted most of the
exposition to random sequences , i.e. , discrete-tim e random processes . Thi s is a
severe restriction indeed, but it makes the presentation much more user-friendly .
The reader ca n expand his or her skills by turning to the monographs that ar e
listed as references. Her e a wealth of refinements and embellishments can be found,
as well as beautiful and deep large deviation results for Brownian motion, random
dynamical systems, Gibb s measures, interactin g particle systems, Brownian mo-
tion among random obstacles, etc. Thes e monographs als o contain an extensiv e
historical overview of the area.
The material presente d her e was taught a s a graduate cours e a t th e Field s
Institute for Research in Mathematical Science s in Toronto, Canada, i n the Fall
of 1998, as part of the 1998-99 program on "Probability and its Applications". I
am grateful t o the staff o f the Fields Institute for the hospitality I enjoyed a s a
visitor. I am grateful to the following colleagues for comments during the course:
Rami Atar, Siva Athreya, Marek Biskup, Jiirgen Gartner, Takash i Hara, Remc o
van der Hofstad, Mi n Kang, Nea l Madras, Ander s Martin-Lof, Georg e O'Brien ,
David Rolls, Tom Salisbury, Gordon Slade, Dean Slonowsky, Jan Swart and Stas
Volkov.
I am indebted to Siva Athreya, Nina Gantert, Jiirgen Gartner, Andreas Greven,
Remco van der Hofstad, Wolfgang Konig and George O'Brien for reading a draft
of the manuscript and for suggesting improvements.
Special thanks go to Marek Biskup, who helped me to prepare the manuscript,
both in terms of content and exposition, and who worked hard on the layout and
on the figures. Mare k was a constant companion in the enterprise of bringing the
job to a good end.
Frank den Hollander
Nijmegen, October 1999
Part A
THEORY