Markov Chains and Invariant Probabilities

ISBN-13: 9783034894081 / Angielski / Miękka / 2012 / 208 str.

Onesimo Hernandez-Lerma; Jean B. Lasserre

Opis

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).

Recenzje

"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

Spis treści

1 Preliminaries.- 1.1 Introduction.- 1.2 Measures and Functions.- 1.3 Weak Topologies.- 1.4 Convergence of Measures.- 1.5 Complements.- 1.6 Notes.- I Markov Chains and Ergodicity.- 2 Markov Chains and Ergodic Theorems.- 2.1 Introduction.- 2.2 Basic Notation and Definitions.- 2.3 Ergodic Theorems.- 2.4 The Ergodicity Property.- 2.5 Pathwise Results.- 2.6 Notes.- 3 Countable Markov Chains.- 3.1 Introduction.- 3.2 Classification of States and Class Properties.- 3.3 Limit Theorems.- 3.4 Notes.- 4 Harris Markov Chains.- 4.1 Introduction.- 4.2 Basic Definitions and Properties.- 4.3 Characterization of Harris recurrence.- 4.4 Sufficient Conditions for P.H.R.- 4.5 Harris and Doeblin Decompositions.- 4.6 Notes.- 5 Markov Chains in Metric Spaces.- 5.1 Introduction.- 5.2 The Limit in Ergodic Theorems.- 5.3 Yosida’s Ergodic Decomposition.- 5.4 Pathwise Results.- 5.5 Proofs.- 5.6 Notes.- 6 Classification of Markov Chains via Occupation Measures.- 6.1 Introduction.- 6.2 A Classification.- 6.3 On the Birkhoff Individual Ergodic Theorem.- 6.4 Notes.- II Further Ergodicity Properties.- 7 Feller Markov Chains.- 7.1 Introduction.- 7.2 Weak-and Strong-Feller Markov Chains.- 7.3 Quasi Feller Chains.- 7.4 Notes.- 8 The Poisson Equation.- 8.1 Introduction.- 8.2 The Poisson Equation.- 8.3 Canonical Pairs.- 8.4 The Cesàro-Averages Approach.- 8.5 The Abelian Approach.- 8.6 Notes.- 9 Strong and Uniform Ergodicity.- 9.1 Introduction.- 9.2 Strong and Uniform Ergodicity.- 9.3 Weak and Weak Uniform Ergodicity.- 9.4 Notes.- III Existence and Approximation of Invariant Probability Measures.- 10 Existence of Invariant Probability Measures.- 10.1 Introduction and Statement of the Problems.- 10.2 Notation and Definitions.- 10.3 Existence Results.- 10.4 Markov Chains in Locally Compact Separable Metric Spaces.- 10.5 Other Existence Results in Locally Compact Separable Metric Spaces.- 10.6 Technical Preliminaries.- 10.7 Proofs.- 10.8 Notes.- 11 Existence and Uniqueness of Fixed Points for Markov Operators.- 11.1 Introduction and Statement of the Problems.- 11.2 Notation and Definitions.- 11.3 Existence Results.- 11.4 Proofs.- 11.5 Notes.- 12 Approximation Procedures for Invariant Probability Measures.- 12.1 Introduction.- 12.2 Statement of the Problem and Preliminaries.- 12.3 An Approximation Scheme.- 12.4 A Moment Approach for a Special Class of Markov Chains.- 12.5 Notes.

Blurb

This book concerns discrete-time homogeneous Markov chains that admit an invariant probability measure. The main objective is to give a systematic, self-contained presentation on some key issues about the ergodic behavior of that class of Markov chains. These issues include, in particular, the various types of convergence of expected and pathwise occupation measures, and ergodic decompositions of the state space. Some of the results presented appear for the first time in book form. A distinguishing feature of the book is the emphasis on the role of expected occupation measures to study the long-run behavior of Markov chains on uncountable spaces.

The intended audience are graduate students and researchers in theoretical and applied probability, operations research, engineering and economics.