Preface to the Second Edition Preface to the First Edition 1 Introduction 1.1 Rare Events and Large Deviations 1.2 The Large Deviation Principle 1.3 Historical Notes and References 2 LDP for Finite Dimensional Spaces 2.1 Combinatorial Techniques for Finite Alphabets 2.1.1 The Method of Types and Sanov's Theorem 2.1.2 Cramer's Theorem for Finite Alphabets in R 2.1.3 Large Deviations for Sampling Without Replacement 2.2 Cramer's Theorem 2.2.1 Cramer's Theorem in R 2.2.2 Cramer's Theorem in Rd 2.3 The Gartner-Ellis Theorem 2.4 Concentration Inequalities 2.4.1 Inequalities for Bounded Martingale Differences 2.4.2 Talagrand's Concentration Inequalities 2.5 Historical Notes and References 3 Applications--The Finite Dimensional Case 3.1 Large Deviations for Finite State Markov Chains 3.1.1 LDP for Additive Functiona of Markov Chains 3.1.2 Sanov's Theorem for the Empirical Measure of Markov Chains 3.1.3 Sanov's Theorem for the Pair Empirical Measure of Markov Chains 3.2 Long Rare Segments in Random Walks 3.3 The Gibbs Conditioning Principle for Finite Alphabets 3.4 The Hypothesis Testing Problem 3.5 Generalized Likelihood Ratio Test for Finite Alphabets 3.6 Rate Distortion Theory 3.7 Moderate Deviations and Exact Asymptotics in Rd 3.8 Historical Notes and References 4 General Principles 4.1 Existence of an LDP and Related Properties 4.1.1 Properties of the LDP 4.1.2 The Existence of an LDP 4.2 Transformations of LDPs 4.2.1 Contraction Principles 4.2.2 Exponential Approximations 4.3 Varadhan's Integral Lemma 4.4 Bryc's Inverse Varadhan Lemma 4.5 LDP in Topological Vector Spaces 4.5.1 A General Upper Bound 4.5.2 Convexity Considerations 4.5.3 Abstract Gartner-Ellis Theorem 4.6 Large Deviations for Projective Limits 4.7 The LDP and Weak Convergence in Metric Spaces 4.8 Historical Notes and References 5 Sample Path Large Deviations 5.1 Sample Path Large Deviations for Random Walks 5.2 Brownian Motion Sample Path Large Deviations
5.3 Multivariate Random Walk and Brownian Sheet 5.4 Performance Analysis of DMPSK Modulation 5.5 Large Exceedances in Rd 5.6 The Freidlin-Wentzell Theory 5.7 The Problem of Diffusion Exit from a Domain 5.8 The Performance of Tracking Loops 5.8.1 An Angular Tracking Loop Analysis 5.8.2 The Analysis of Range Tracking Loops 5.9 Historical Notes and References 6 The LDP for Abstract Empirical Measures 6.1 Cramer's Theorem in Polish Spaces 6.2 Sanov's Theorem 6.3 LDP for the Empirical Measure---The Uniform Markov Case 6.4 Mixing Conditions and LDP 6.4.1 LDP for the Empirical Mean in Rd 6.4.2 Empirical Measure LDP for Mixing Processes 6.5 LDP for Empirical Measures of Markov Chains 6.5.1 LDP for Occupation Times 6.5.2 LDP for the k-Empirical Measures 6.5.3 Process Level LDP for Markov Chains 6.6 A Weak Convergence Approach to Large Deviations 6.7 Historical Notes and References 7 Applications of Empirical Measures LDP 7.1 Universal Hypothesis Testing 7.1.1 A General Statement of Test Optimality 7.1.2 Independent and Identically Distributed Observations 7.2 Sampling Without Replacement 7.3 The Gibbs Conditioning Principle 7.3.1 The Non-Interacting Case 7.3.2 The Interacting Case 7.3.3 Refinements of the Gibbs Conditioning Principle 7.4 Historical Notes and References Appendix A Convex Analysis Considerations in Rd B Topological Preliminaries B.1 Generalities B.2 Topological Vector Spaces and Weak Topologies B.3 Banach and Polish Spaces B.4 Mazur's Theorem C Integration and Function Spaces C.1 Additive Set Functions C.2 Integration and Spaces of Functions D Probability Measures on Polish Spaces D.1 Generalities D.2 Weak Topology D.3 Product Space and Relative Entropy Decompositions E Stochastic Analysis Bibliography General Conventions Index of Notation
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