Markov Chains and Invariant Probabilities

by Hernandez-Lerma, O.; Lasserre, Jean-Bernard

• ISBN13: 9783764370008
• ISBN10: 3764370009
• Format: Hardcover
• Copyright: 2003-06-01
• Publisher: Birkhauser

Summary

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.

Table of Contents

Acknowledgements	ix
Preface	xi
Abbreviations and List of Symbols	xv
Preliminaries	1	(18)
Introduction	1	(1)
Measures and Functions	1	(3)
Weak Topologies	4	(2)
Convergence of Measures	6	(7)
Complements	13	(4)
Notes	17	(2)
I Markov Chains and Ergodicity	19	(74)
Markov Chains and Ergodic Theorems	21	(20)
Introduction	21	(1)
Basic Notation and Definitions	22	(6)
Ergodic Theorems	28	(5)
The Ergodicity Property	33	(3)
Pathwise Results	36	(3)
Notes	39	(2)
Countable Markov Chains	41	(6)
Introduction	41	(1)
Classification of States and Class Properties	41	(3)
Limit Theorems	44	(2)
Notes	46	(1)
Harris Markov Chains	47	(16)
Introduction	47	(1)
Basic Definitions and Properties	47	(4)
Characterization of Harris recurrence	51	(5)
Sufficient Conditions for P.H.R.	56	(4)
Harris and Doeblin Decompositions	60	(1)
Notes	61	(2)
Markov Chains in Metric Spaces	63	(20)
Introduction	63	(1)
The Limit in Ergodic Theorems	64	(4)
Yosida's Ergodic Decomposition	68	(5)
Pathwise Results	73	(1)
Proofs	74	(8)
Notes	82	(1)
Classification of Markov Chains via Occupation Measures	83	(10)
Introduction	83	(1)
A Classification	84	(4)
On the Birkhoff Individual Ergodic Theorem	88	(4)
Notes	92	(1)
II Further Ergodicity Properties	93	(40)
Feller Markov Chains	95	(8)
Introduction	95	(1)
Weak- and Strong-Feller Markov Chains	96	(3)
Quasi Feller Chains	99	(3)
Notes	102	(1)
The Poisson Equation	103	(18)
Introduction	103	(1)
The Poisson Equation	103	(2)
Canonical Pairs	105	(5)
The Cesaro-Averages Approach	110	(4)
The Abelian Approach	114	(5)
Notes	119	(2)
Strong and Uniform Ergodicity	121	(12)
Introduction	121	(1)
Strong and Uniform Ergodicity	122	(5)
Weak and Weak Uniform Ergodicity	127	(4)
Notes	131	(2)
III Existence and Approximation of Invariant Probability Measures	133	(60)
Existence of Invariant Probability Measures	135	(22)
Introduction and Statement of the Problems	135	(1)
Notation and Definitions	136	(2)
Existence Results	138	(5)
Markov Chains in Locally Compact Separable Metric Spaces	143	(2)
Other Existence Results in Locally Compact Separable Metric Spaces	145	(2)
Technical Preliminaries	147	(2)
Proofs	149	(6)
Notes	155	(2)
Existence and Uniqueness of Fixed Points for Markov Operators	157	(18)
Introduction and Statement of the Problems	157	(1)
Notation and Definitions	158	(2)
Existence Results	160	(7)
Proofs	167	(7)
Notes	174	(1)
Approximation Procedures for Invariant Probability Measures	175	(18)
Introduction	175	(1)
Statement of the Problem and Preliminaries	176	(2)
An Approximation Scheme	178	(5)
A Moment Approach for a Special Class of Markov Chains	183	(7)
Notes	190	(3)
Bibliography	193	(10)
Index	203