A Weak Convergence Approach to the Theory of Large Deviations

by Dupuis, Paul; Ellis, Richard S.

• ISBN13: 9780471076728
• ISBN10: 0471076724
• Edition: 1st
• Format: Hardcover
• Copyright: 1997-02-27
• Publisher: Wiley-Interscience

Summary

Applies the well-developed tools of the theory of weak convergence of probability measures to large deviation analysis-a consistent new approach The theory of large deviations, one of the most dynamic topics in probability today, studies rare events in stochastic systems. The nonlinear nature of the theory contributes both to its richness and difficulty. This innovative text demonstrates how to employ the well-established linear techniques of weak convergence theory to prove large deviation results. Beginning with a step-by-step development of the approach, the book skillfully guides readers through models of increasing complexity covering a wide variety of random variable-level and process-level problems. Representation formulas for large deviation-type expectations are a key tool and are developed systematically for discrete-time problems. Accessible to anyone who has a knowledge of measure theory and measure-theoretic probability, A Weak Convergence Approach to the Theory of Large Deviations is important reading for both students and researchers.

Author Biography

PAUL DUPUIS is a professor in the Division of Applied Mathematics at Brown University in Providence, Rhode Island. RICHARD S. ELLIS is a professor in the Department of Mathematics and Statistics at the University of Massachusetts at Amherst.

Table of Contents

1. Formulation of Large Deviation Theory in Terms of the Laplace Principle	1	(47)
1.1. Introduction	1	(3)
1.2. Equivalent Formulation of the Large Deviation Principle	4	(12)
1.3. Basic Results in the Theory	16	(16)
1.4. Properties of the Relative Entropy	32	(9)
1.5. Stochastic Control Theory and Dynamic Programming	41	(7)
2. First Example: Sanov's Theorem	48	(17)
2.1. Introduction	48	(1)
2.2. Statement of Sanov's Theorem	49	(2)
2.3. The Representation Formula	51	(7)
2.4. Proof of the Laplace Principle Lower Bound	58	(1)
2.5. Proof of the Laplace Principle Upper Bound	58	(7)
3. Second Example: Mogulskii's Theorem	65	(27)
3.1. Introduction	65	(2)
3.2. The Representation Formula	67	(7)
3.3. Proof of the Laplace Principle Upper Bound and Identification of the Rate Function	74	(7)
3.4. Statement of Mogulskii's Theorem and Completion of the Proof	81	(5)
3.5. Cramer's Theorem	86	(2)
3.6. Comments on the Proofs	88	(4)
4. Representation Formulas for Other Stochastic Processes	92	(35)
4.1. Introduction	92	(2)
4.2. The Representation Formula for the Empirical Measures of a Markov Chain	94	(4)
4.3. The Representation Formula for a Random Walk Model	98	(4)
4.4. The Representation Formula for a Random Walk Model with State-Dependent Noise	102	(5)
4.5. Extensions to Unbounded Functions	107	(4)
4.6. Representation Formulas for Continuous-Time Markov Processes	111	(16)
4.6.1. Introduction	111	(2)
4.6.2. Formal Derivation of Representation Formulas for Continuous-Time Markov Processes	113	(6)
4.6.3. Examples of Continuous-Time Representation Formulas	119	(4)
4.6.4. Remarks on the Proofs of the Representation Formulas	123	(4)
5. Compactness and Limit Properties for the Random Walk Model	127	(22)
5.1. Introduction	127	(1)
5.2. Definitions and a Representation Formula	128	(3)
5.3. Compactness and Limit Properties	131	(16)
5.4. Weaker Version of Condition 5.3.1	147	(2)
6. Laplace Principle for the Random Walk Model with Continuous Statistics	149	(67)
6.1. Introduction	149	(2)
6.2. Proof of the Laplace Principle Upper Bound and Identification of the Rate Function	151	(12)
6.3. Statement of the Laplace Principle	163	(8)
6.4. Strategy for the Proof of the Laplace Principle Lower Bound	171	(6)
6.5. Proof of the Laplace Principle Lower Bound Under Conditions 6.2.1 and 6.3.1	177	(8)
6.6. Proof of the Laplace Principle Lower Bound Under Conditions 6.2.1 and 6.3.2	185	(21)
6.7. Extension of Theorem 6.3.3 To Be Applied in Chapter 10	206	(10)
7. Laplace Principle for the Random Walk Model with Discontinuous Statistics	216	(59)
7.1. Introduction	216	(2)
7.2. Statement of the Laplace Principle	218	(4)
7.3. Laplace Principle for the Final Position Vectors and One-domensional Examples	222	(5)
7.4. Proof of the Laplace Principle Upper Bound	227	(21)
7.5. Proof of the Laplace Principle Lower Bound	248	(22)
7.6. Compactness of the Level Sets of Ix	270	(5)
8. Laplace Principle for the Empirical Measures of a Markov Chain	275	(45)
8.1 Introduction	275	(2)
8.2. Compactness and Limit Properties of Controls and Controlled Processes	277	(14)
8.3. Proof of the Laplace Principle Upper Bound and Identification of the Rate Function	291	(7)
8.4. Statement of the Laplace Principle	298	(2)
8.5. Properties of the Rate Function	300	(5)
8.6. Proofs of the Laplace Principle Lower Bounds	305	(15)
9. Extensions of the Laplace Principle for the Empirical Measures of a Markov Chain	320	(30)
9.1. Introduction	320	(2)
9.2. Laplace Principle for the Empirical Measures of a Markov Chain with Discontinuous Statistics	322	(10)
9.3. Laplace Limit for the Empirical Measures of a Markov Chain in the T-Topology	332	(18)
10. Laplace Principle for Continuous-Time Markov Processes with Continuous Statistics	350	(21)
10.1 Introduction	350	(1)
10.2 Statement of the Laplace Principle	350	(7)
10.3 Proof pf the Laplace Principle	357	(14)
Appendix A. Background Material	371	(29)
A.1. Introduction	371	(1)
A.2. Measure Theory	371	(2)
A.3. Weak Convergence of Probability Measures	373	(12)
A.4. Probability Theory	385	(4)
A.5. Stochastic Kernels	389	(8)
A.6. Analysis	397	(3)
Appendix B. Deriving the Representation Formulas via Measure Theory	400	(5)
B.1. Introduction	400	(1)
B.2. Measure -Theoretic Proof of the Representation Formula for Sanov's Theorem	400	(3)
B.3. Discussion	403	(3)
Appendix C. Proofs of a Number of Results	405	(26)
C.1. Introduction	405	(1)
C.2. Proof of the Donsker-Varadhan Variational Formula for the Relative Entropy	405	(3)
C.3. Proof of the Chain Rule and Part (f) of Lemma 1.4.3	408	(4)
C.4. Proof of Part (g) of Lemma 1.4.3	412	(4)
C.5. Proof of Part (f) of Lemma 6.2.3	416	(4)
C.6. Proof of Part (g) of Lemma 6.2.3	420	(3)
C.7. Proof of Proposition 6.3.4	423	(4)
C.8. Continuity Property of Cramer Functions	427	(4)
Appendix D. Convex Functions	431	(18)
D.1. Introduction	431	(1)
D.2. Background Material on Convex Functions	431	(5)
D.3. Theorem on the Legendre-Fenchel Transform of Compositions of Convex Functions	436	(9)
D.4. Three Examples	445	(4)
Appendix E. Proof of Theorem 5.3.5 When Condition 5.4.1 Replaces Condition 5.3.1	449	(9)
E.1. Introduction	449	(1)
E.2. Proofs of Results	449	(9)
Bibliography	458	(5)
Notation Index	463	(4)
Author Index	467	(2)
Subject Index	469