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