A Weak Convergence Approach to the Theory of Large Deviations

by Paul Dupuis and Richard S. Ellis

 

 

 

Wiley Series in Probability and Statistics

John Wiley & Sons, Inc., New York, 1997

ISBN 0-471-07672-4, xvii + 479 pp.

 

For information and ordering, call 800-225-5945.

 

 

This book presents a comprehensive new approach to the theory of large deviations based on the theory of weak convergence of  probability measures.  The first step in the approach is to replace the analysis of the asymptotic behavior of the normalized logarithms of probabilities by the analysis of normalized logarithms of expectations involving continuous functions.  We refer to the latter as the Laplace principle, which is shown to be equivalent to a large deviation principle with the same rate function.  The second step is to represent the normalized logarithms of the expectations as variational formulas that are then interpreted, in a natural way, as the minimal cost functions of associated stochastic optimal control problems.  These minimal cost functions have a form to which the theory of weak convergence of probability measures can be applied.  We consider the introduction of representation formulas to be one of our main contributions.  They allow us to replace the exponential estimates of standard large deviation approaches by law of large numbers-type estimates, which are obtained by the theory of weak convergence. 

This book is easily accessible to anyone who has taken courses measure theory and measure-theoretic probability.  No background in control theory is assumed or required.  Indeed, control theory is used mainly to provide intuition and a convenient terminology. Much of the material in the book is being published here for the first time.

Chapter 1 of the book introduces the Laplace principle and then proves a number of general results in large deviation theory from the Laplace principle perspective.  This chapter introduces a main theme that runs through the entire work: namely, the central role played by relative entropy in large deviation theory.  The remainder of the book studies two classes of applications: empirical measures of Markov chains and random walk models.  Important special cases of these two classes are treated in Chapters 2 and 3.  Chapter 4 develops the representation formulas that will be used in subsequent chapters to prove the Laplace principle.  Chapters 5 through 7 and Chapter 10 are devoted to random walk models and their continuous-time analogues, while Chapters 8 and 9 treat the empirical measure problem under various assumptions.  In all cases the weak convergence approach enables us not only to reproduce known results, often under weaker conditions, but also to treat new problems.     

The power of the weak convergence approach can be seen in the study of Markov processes with discontinuous statistics, which arise when modeling queueing systems, in the study of computer  and communication networks, in the analysis of controlled diffusions with discontinuous feedback controls, and in other areas.  Markov processes with discontinuous statistics are characterized by the fact that the components of their generators do not depend smoothly on the spatial parameter.  As a result of this nonsmooth dependence, classical methods for studying these processes are not applicable.  Two examples in which the weak convergence approach is particularly useful are given in Chapters 7 and 9 of the book, where we treat a random walk model with discontinuous statistics and the empirical measures of a Markov chain with discontinuous statistics.  Markov processes with continuous statistics are analyzed in Chapters 6, 9, and 10.

 

CONTENTS

Chapter 1.  Formulation of Large Deviation Theory in Terms of the Laplace Principle

Chapter 2.  First Example: Sanov's Theorem

Chapter 3.  Second Example: Mogulskii's Theorem

Chapter 4.  Representation Formulas for Other Stochastic Processes

Chapter 5.  Compactness and Limit Properties for the Random Walk Model

Chapter 6.  Laplace Principle for the Random Walk Model with Continuous Statistics

Chapter 7.  Laplace Principle for the Random Walk Model with Discontinuous Statistics

Chapter 8.  Laplace Principle for the Empirical Measures of a Markov Chain

Chapter 9.  Extensions of the Laplace Principle for the Empirical Measures of a Markov Chain

Chapter 10. Laplace Principle for Continuous–Time Markov Processes with Continuous Statistics

Appendix A. Background Material

Appendix B. Deriving the Representation Formulas via Measure Theory

Appendix C. Proofs of a Number of Results

Appendix D. Convex Functions

Appendix E. Proof of Theorem 5.3.5 When Condition 5.4.1 Replaces Condition 5.3.1