Large Deviations Techniques and Applications

Transcription

1 Amir Dembo Ofer Zeitouni Large Deviations Techniques and Applications Second Edition With 29 Figures Springer

2 Contents Preface to the Second Edition Preface to the First Edition vii ix 1 Introduction Rare Events and Large Deviations The Large Deviation Principle Historical Notes and References 9 2 LDP for Finite Dimensional Spaces Combinatorial Techniques for Finite Alphabets The Method of Types and Sanov's Theorem Cramer's Theorem for Finite Alphabets in IR Large Deviations for Sampling Without Replacement Cramer's Theorem Cramer's Theorem in IR Cramer's Theorem in M d The Gartner-Ellis Theorem Concentration Inequalities Inequalities for Bounded Martingale Differences Talagrand's Concentration Inequalities Historical Notes and References 68 xin

3 xiv 3 Applications The Finite Dimensional Case Large Deviations for Finite State Markov Chains LDP for Additive Functionals of Markov Chains Sanov's Theorem for the Empirical Measure of Markov Chains Sanov's Theorem for the Pair Empirical Measure of Markov Chains Long Rare Segments in Random Walks The Gibbs Conditioning Principle for Finite Alphabets The Hypothesis Testing Problem Generalized Likelihood Ratio Test for Finite Alphabets Rate Distortion Theory Moderate Deviations and Exact Asymptotics in IR d Historical Notes and References General Principles Existence of an LDP and Related Properties Properties of the LDP The Existence of an LDP Transformations of LDPs Contraction Principles Exponential Approximations Varadhan's Integral Lemma Bryc's Inverse Varadhan Lemma LDP in Topological Vector Spaces A General Upper Bound Convexity Considerations Abstract Gartner-Ellis Theorem Large Deviations for Projective Limits The LDP and Weak Convergence in Metric Spaces Historical Notes and References.. 173

4 XV 5 Sample Path Large Deviations Sample Path Large Deviations for Random Walks Brownian Motion Sample Path Large Deviations Multivariate Random Walk and Brownian Sheet Performance Analysis of DMPSK Modulation Large Exceedances in IR d The Freidlin-Wentzell Theory The Problem of Diffusion Exit from a Domain The Performance of Tracking Loops An Angular Tracking Loop Analysis The Analysis of Range Tracking Loops Historical Notes and References The LDP for Abstract Empirical Measures Cramer's Theorem in Polish Spaces Sanov's Theorem LDP for the Empirical Measure The Uniform Markov Case Mixing Conditions and LDP LDP for the Empirical Mean in K d Empirical Measure LDP for Mixing Processes LDP for Empirical Measures of Markov Chains LDP for Occupation Times LDP for the fc-empirical Measures Process Level LDP for Markov Chains A Weak Convergence Approach to Large Deviations Historical Notes and References. ^ Applications of Empirical Measures LDP Universal Hypothesis Testing A General Statement of Test Optimality Independent and Identically Distributed Observations Sampling Without Replacement 318

5 xvi 7.3 The Gibbs Conditioning Principle The Non-Interacting Case The Interacting Case Refinements of the Gibbs Conditioning Principle Historical Notes and References 338 Appendix 341 A Convex Analysis Considerations in lr d 341 B Topological Preliminaries 343 B.I Generalities 343 B.2 Topological Vector Spaces and Weak Topologies B.3 Banach and Polish Spaces 347 B.4 Mazur's Theorem 349 C Integration and Function Spaces 350 C.I Additive Set Functions 350 C.2 Integration and Spaces of Functions 352 D Probability Measures on Polish Spaces 354 D.I Generalities 354 D.2 Weak Topology 355 D.3 Product Space and Relative Entropy Decompositions 357 E Stochastic Analysis 359 Bibliography 363 General Conventions 385 Index of Notation 387 Index 391