Markov Chains

Description

This book covers the classical theory of Markov chains on general state-spaces as well as many recent developments. The theoretical results are illustrated by simple examples, many of which are taken from Markov Chain Monte Carlo methods. The book is self-contained, while all the results are carefully and concisely proven. Bibliographical notes are added at the end of each chapter to provide an overview of the literature. Part I lays the foundations of the theory of Markov chain on general states-space. Part II covers the basic theory of irreducible Markov chains on general states-space, relying heavily on regeneration techniques. These two parts can serve as a text on general state-space applied Markov chain theory. Although the choice of topics is quite different from what is usually covered, where most of the emphasis is put on countable state space, a graduate student should be able to read almost all these developments without any mathematical background deeper than that needed to study countable state space (very little measure theory is required). Part III covers advanced topics on the theory of irreducible Markov chains. The emphasis is on geometric and subgeometric convergence rates and also on computable bounds. Some results appeared for a first time in a book and others are original. Part IV are selected topics on Markov chains, covering mostly hot recent developments. Front Matter ....Pages i-xviii Front Matter ....Pages 1-1 Markov Chains: Basic Definitions (Randal Douc, Eric Moulines, Pierre Priouret, Philippe Soulier)....Pages 3-25 Examples of Markov Chains (Randal Douc, Eric Moulines, Pierre Priouret, Philippe Soulier)....Pages 27-52 Stopping Times and the Strong Markov Property (Randal Douc, Eric Moulines, Pierre Priouret, Philippe Soulier)....Pages 53-74 Martingales, Harmonic Functions and Poisson–Dirichlet Problems (Randal Douc, Eric Moulines, Pierre Priouret, Philippe Soulier)....Pages 75-96 Ergodic Theory for Markov Chains (Randal Douc, Eric Moulines, Pierre Priouret, Philippe Soulier)....Pages 97-115 Front Matter ....Pages 117-117 Atomic Chains (Randal Douc, Eric Moulines, Pierre Priouret, Philippe Soulier)....Pages 119-144 Markov Chains on a Discrete State Space (Randal Douc, Eric Moulines, Pierre Priouret, Philippe Soulier)....Pages 145-164 Convergence of Atomic Markov Chains (Randal Douc, Eric Moulines, Pierre Priouret, Philippe Soulier)....Pages 165-189 Small Sets, Irreducibility, and Aperiodicity (Randal Douc, Eric Moulines, Pierre Priouret, Philippe Soulier)....Pages 191-220 Transience, Recurrence, and Harris Recurrence (Randal Douc, Eric Moulines, Pierre Priouret, Philippe Soulier)....Pages 221-239 Splitting Construction and Invariant Measures (Randal Douc, Eric Moulines, Pierre Priouret, Philippe Soulier)....Pages 241-264 Feller and T-Kernels (Randal Douc, Eric Moulines, Pierre Priouret, Philippe Soulier)....Pages 265-286 Front Matter ....Pages 287-287 Rates of Convergence for Atomic Markov Chains (Randal Douc, Eric Moulines, Pierre Priouret, Philippe Soulier)....Pages 289-312 Geometric Recurrence and Regularity (Randal Douc, Eric Moulines, Pierre Priouret, Philippe Soulier)....Pages 313-337 Geometric Rates of Convergence (Randal Douc, Eric Moulines, Pierre Priouret, Philippe Soulier)....Pages 339-359 (f, r)-Recurrence and Regularity (Randal Douc, Eric Moulines, Pierre Priouret, Philippe Soulier)....Pages 361-383 Subgeometric Rates of Convergence (Randal Douc, Eric Moulines, Pierre Priouret, Philippe Soulier)....Pages 385-400 Uniform and V-Geometric Ergodicity by Operator Methods (Randal Douc, Eric Moulines, Pierre Priouret, Philippe Soulier)....Pages 401-419 Coupling for Irreducible Kernels (Randal Douc, Eric Moulines, Pierre Priouret, Philippe Soulier)....Pages 421-452 Front Matter ....Pages 453-453 Convergence in the Wasserstein Distance (Randal Douc, Eric Moulines, Pierre Priouret, Philippe Soulier)....Pages 455-488 Central Limit Theorems (Randal Douc, Eric Moulines, Pierre Priouret, Philippe Soulier)....Pages 489-522 Spectral Theory (Randal Douc, Eric Moulines, Pierre Priouret, Philippe Soulier)....Pages 523-574 Concentration Inequalities (Randal Douc, Eric Moulines, Pierre Priouret, Philippe Soulier)....Pages 575-601 Back Matter ....Pages 603-757