Synopses & Reviews
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.
Review
"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
Synopsis
Includes bibliographical references (p. [193]-201) and index.
Synopsis
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).
Table of Contents
Preliminaries .- Markov Chains and Ergodic Theorems .- Countable Markov Chains .- Harris Markov Chains .- Markov Chains in Metric Spaces .- Classification of Markov Chains via Occupation Measures .- Feller Markov Chains .- The Poisson Equation .- Strong and Uniform Ergodicity .- Existence of Invariant Probability Measures .- Existence and Uniqueness of Fixed Points for Markov Operators .- Approximation Procedures for Invariant Probability Measures