Large Deviations (Fields Institute Monographs)

This volume offers an introduction to large deviations. It is divided into two parts: theory and applications. Basic large deviation theorems are presented for i.i.d. sequences, Markov sequences, and sequences with moderate dependence. The rate function is computed explicitly. The theory is explained without too much emphasis on technicalities. Also included is an outline of general definitions and theorems. The goal is to expose the unified theme that gives large deviation theory its overall structure, which can be made to work in many concrete cases. The section on applications focuses on recent work in statistical physics and random media. This book contains 60 exercises (with solutions) that should elucidate the content and engage the reader. Prerequisites for the book are a strong background in probability and analysis and some knowledge of statistical physics. It would make an excellent textbook for a special topics course in large deviations. This book is an introduction to the theory and applications of large deviations, a branch of probability theory that describes the probability of rare events in terms of variational problems. By focusing the theory, in Part A of the book, on random sequences, the author succeeds in conveying the main ideas behind large deviations without a need for technicalities, thus providing a concise and accessible entry to this challenging and captivating subject. The selection of modern applications, described in Part B of the book, offers a good sample of what large deviation theory is able to achieve in various different contexts: statistical hypothesis testing, random walk in random environment, heat conduction with random sources and sinks, polymer chains, and interacting diffusions. With its 60 exercises and solutions, this book can be used as a text for graduate students who have had some exposure to mathematical analysis and measure-theoretic probability. Contents Preface Part A. THEORY I. LARGE DEVIATIONS FOR I.I.D. SEQUENCES: PART 1 II. LARGE DEVIATIONS FOR I.I.D. SEQUENCES: PART 2 III. GENERAL THEORY IV. LARGE DEVIATIONS FOR MARKOV SEQUENCES V. LARGE DEVIATIONS FOR DEPENDENT SEQUENCES Part B. APPLICATIONS VI. STATISTICAL HYPOTHESIS TESTING VII. RANDOM WALK IN RANDOM ENVIRONMENT VIII. HEAT CONDUCTION WITH RANDOM SOURCES ANDSINKS IX. POLYMER CHAINS X. INTERACTING DIFFUSIONS Appendix: Solutions to the Exercises Bibliography Index Glossary of Symbols Errata Errata for the Reprinted Edition Offers an introduction to large deviations. This book is divided into two parts: theory and applications. It presents basic large deviation theorems for i i d sequences, Markov sequences, and sequences with moderate dependence. It also includes an outline of general definitions and theorems.