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Markov Chains

  • Randal Douc
  • Eric Moulines
  • Pierre Priouret
  • Philippe Soulier

Table of contents

  1. Front Matter
    Pages i-xviii
  2. Foundations

    1. Front Matter
      Pages 1-1
    2. Randal Douc, Eric Moulines, Pierre Priouret, Philippe Soulier
      Pages 3-25
    3. Randal Douc, Eric Moulines, Pierre Priouret, Philippe Soulier
      Pages 27-52
    4. Randal Douc, Eric Moulines, Pierre Priouret, Philippe Soulier
      Pages 53-74
    5. Randal Douc, Eric Moulines, Pierre Priouret, Philippe Soulier
      Pages 75-96
    6. Randal Douc, Eric Moulines, Pierre Priouret, Philippe Soulier
      Pages 97-115
  3. Irreducible Chains: Basics

    1. Front Matter
      Pages 117-117
    2. Randal Douc, Eric Moulines, Pierre Priouret, Philippe Soulier
      Pages 119-144
    3. Randal Douc, Eric Moulines, Pierre Priouret, Philippe Soulier
      Pages 145-164
    4. Randal Douc, Eric Moulines, Pierre Priouret, Philippe Soulier
      Pages 165-189
    5. Randal Douc, Eric Moulines, Pierre Priouret, Philippe Soulier
      Pages 191-220
    6. Randal Douc, Eric Moulines, Pierre Priouret, Philippe Soulier
      Pages 221-239
    7. Randal Douc, Eric Moulines, Pierre Priouret, Philippe Soulier
      Pages 241-264
    8. Randal Douc, Eric Moulines, Pierre Priouret, Philippe Soulier
      Pages 265-286
  4. Irreducible Chains: Advanced Topics

    1. Front Matter
      Pages 287-287
    2. Randal Douc, Eric Moulines, Pierre Priouret, Philippe Soulier
      Pages 289-312
    3. Randal Douc, Eric Moulines, Pierre Priouret, Philippe Soulier
      Pages 313-337
    4. Randal Douc, Eric Moulines, Pierre Priouret, Philippe Soulier
      Pages 339-359
    5. Randal Douc, Eric Moulines, Pierre Priouret, Philippe Soulier
      Pages 361-383
    6. Randal Douc, Eric Moulines, Pierre Priouret, Philippe Soulier
      Pages 385-400
    7. Randal Douc, Eric Moulines, Pierre Priouret, Philippe Soulier
      Pages 401-419
    8. Randal Douc, Eric Moulines, Pierre Priouret, Philippe Soulier
      Pages 421-452
  5. Selected Topics

    1. Front Matter
      Pages 453-453
    2. Randal Douc, Eric Moulines, Pierre Priouret, Philippe Soulier
      Pages 455-488
    3. Randal Douc, Eric Moulines, Pierre Priouret, Philippe Soulier
      Pages 489-522
    4. Randal Douc, Eric Moulines, Pierre Priouret, Philippe Soulier
      Pages 523-574
    5. Randal Douc, Eric Moulines, Pierre Priouret, Philippe Soulier
      Pages 575-601
  6. Back Matter
    Pages 603-757

About this book

Introduction

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 state-spaces. Part II covers the basic theory of irreducible Markov chains starting from the definition of small and petite sets, the characterization of recurrence and transience and culminating in the Harris theorem. Most of the results rely on the splitting technique which allows to reduce the theory of irreducible to a Markov chain with an atom. These two parts can serve as a text on Markov chain theory on general state-spaces. Although the choice of topics is quite different from what is usually covered in a classical Markov chain course, where most of the emphasis is put on countable state space, a graduate student should be able to read almost all of these developments without any mathematical background deeper than that needed to study countable state space (very little measure theory is required). 

Part III deals with advanced topics on the theory of irreducible Markov chains, covering geometric and subgeometric convergence rates. Special attention is given to obtaining computable convergence bounds using Foster-Lyapunov drift conditions and minorization techniques. 

Part IV presents selected topics on Markov chains, covering mostly hot recent developments. It represents a biased selection of topics, reflecting the authors own research inclinations. This includes quantitative bounds of convergence in Wasserstein distances, spectral theory of Markov operators, central limit theorems for additive functionals and concentration inequalities.

Some of the results in Parts III and IV appear for the first time in book form and some are original.

Keywords

Markov Chains Geometric Ergodicity Ergodic Theory Markov Property Potential Theory Operator Theory Limit Theorems Martingales

Authors and affiliations

  • Randal Douc
    • 1
  • Eric Moulines
    • 2
  • Pierre Priouret
    • 3
  • Philippe Soulier
    • 4
  1. 1.Département CITITelecom SudParisÉvryFrance
  2. 2.Centre de Mathématiques AppliquéesEcole PolytechniquePalaiseauFrance
  3. 3.Université Pierre et Marie CurieParisFrance
  4. 4.Université Paris NanterreNanterreFrance

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-319-97704-1
  • Copyright Information Springer Nature Switzerland AG 2018
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-319-97703-4
  • Online ISBN 978-3-319-97704-1
  • Series Print ISSN 1431-8598
  • Series Online ISSN 2197-1773
  • Buy this book on publisher's site