Markov Chains and Invariant Probabilities

1. Front Matter

   Pages i-xvi

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2. Preliminaries

   1. Preliminaries

      • Onésimo Herná-Lerma, Jean Bernard Lasserre

      Pages 1-18

3. Markov Chains and Ergodicity

   1. Front Matter

      Pages 19-19

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   2. Markov Chains and Ergodic Theorems

      • Onésimo Herná-Lerma, Jean Bernard Lasserre

      Pages 21-39

   3. Countable Markov Chains

      • Onésimo Herná-Lerma, Jean Bernard Lasserre

      Pages 41-46

   4. Harris Markov Chains

      • Onésimo Herná-Lerma, Jean Bernard Lasserre

      Pages 47-61

   5. Markov Chains in Metric Spaces

      • Onésimo Herná-Lerma, Jean Bernard Lasserre

      Pages 63-82

   6. Classification of Markov Chains via Occupation Measures

      • Onésimo Herná-Lerma, Jean Bernard Lasserre

      Pages 83-92

4. Further Ergodicity Properties

   1. Front Matter

      Pages 93-93

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   2. Feller Markov Chains

      • Onésimo Herná-Lerma, Jean Bernard Lasserre

      Pages 95-102

   3. The Poisson Equation

      • Onésimo Herná-Lerma, Jean Bernard Lasserre

      Pages 103-120

   4. Strong and Uniform Ergodicity

      • Onésimo Herná-Lerma, Jean Bernard Lasserre

      Pages 121-131

5. Existence and Approximation of Invariant Probability Measures

   1. Front Matter

      Pages 133-133

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   2. Existence and Uniqueness of Invariant Probability Measures

      • Onésimo Herná-Lerma, Jean Bernard Lasserre

      Pages 135-155

   3. Existence and Uniqueness of Fixed Points for Markov Operators

      • Onésimo Herná-Lerma, Jean Bernard Lasserre

      Pages 157-174

   4. Approximation Procedures for Invariant Probability Measures

      • Onésimo Herná-Lerma, Jean Bernard Lasserre

      Pages 175-191

6. Back Matter

   Pages 193-208

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This book is about discrete-time, time-homogeneous, Markov chains (Mes) and their ergodic behavior. To this end, most of the material is in fact about stable Mes, by which we mean Mes that admit an invariant probability measure. To state this more precisely and give an overview of the questions we shall be dealing with, we will first introduce some notation and terminology. Let (X,B) be a measurable space, and consider a X-valued Markov chain ~. = {~k' k = 0, 1, ... } with transition probability function (t.pJ.) P(x, B), i.e., P(x, B) := Prob (~k+1 E B I ~k = x) for each x E X, B E B, and k = 0,1, .... The Me ~. is said to be stable if there exists a probability measure (p.m.) /.l on B such that (*) VB EB. /.l(B) = Ix /.l(dx) P(x, B) If (*) holds then /.l is called an invariant p.m. for the Me ~. (or the t.p.f. P).

• Markov chain
• ergodicity
• mathematical methods in physics
• probability measure
• probability theory

"It should be stressed that an important part of the results presented is due to the authors. . . . In the reviewer's opinion, this is an elegant and most welcome addition to the rich literature of Markov processes."

--MathSciNet

• Departamento de Matemáticas, CINVESTAV-IPN, México, México

  Onésimo Hernández-Lerma

• LAAS-CNRS, France

  Jean Bernard Lasserre

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