From: Paul Dupuis (dupuis@cfm.brown.edu)
	 Richard S. Ellis (rsellis@math.umass.edu) 
Subject: Announcement of Recently Published Book

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, please contact Beth Schacht at 
John Wiley & Sons: bschacht@wiley.com or 212-850-6678.


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 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 in
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