Invariant probability measures for a class of Feller Markov chains
type: Item
instance of: scholarly article
title: Invariant probability measures for a class of Feller Markov chains
zbMATH Open document ID: 0970.60082
DOI: 10.1016/S0167-7152(00)00075-4
published in: Statistics \& Probability Letters
publication date: 2001-10-16T00:00:00Z
review text: In the theory of Markov chains, the problem of obtaining suitable conditions for the existence of an invariant probability measure is of great importance. This paper considers Markov chains defined on locally compact separable metric spaces and satisfying the Feller property. Feller-Markov chains, for which the associated transition probability kernel maps the space of continuous bounded functions into itself, address many problems in several areas of applications. The main purpose of the present paper is related to a result of S. Meyn and R. Tweedie (1993), which developed a Foster-type criterion to guarantee the existence of an invariant measure for Feller-Markov chains. The condition obtained by Meyn and Tweedie is proved to be sufficient, and their criterion is powerful in practice since it is easier to be checked. The authors succeed to prove also the necessity of the Foster-type criterion of Meyn and Tweedie, under a new assumption which generalizes \(T\)-chain irreducibility conditions. Under this new assumption, the Foster-type criterion is shown to be equivalent to the existence of an invariant probability measure for Feller-Markov chains, which is also equivalent to the existence of a non-singular invariant probability measure.
reviewed by: Neculai Curteanu
Mathematics Subject Classification ID: 60J10
Mathematics Subject Classification ID: 93E15
zbMATH DE Number: 1548155
zbMATH Keywords: Markov chains
zbMATH Keywords: Feller-Markov chains
zbMATH Keywords: Foster-type criterion
zbMATH Keywords: invariant measure
zbMATH Keywords: invariant probability measure
author: Oswaldo Luiz V. Costa
author: F. Dufour
MaRDI profile type: Publication
cites work: Stability of Piecewise-Deterministic Markov Processes
cites work: Ergodic theorems and ergodic decomposition for Markov chains
cites work: Invariant probabilities for Markov chains on a metric space
cites work: Markov chains and stochastic stability
cites work: General Irreducible Markov Chains and Non-Negative Operators
cites work: Q5531981
full work available at URL: s0167-7152(00)00075-4
OpenAlex ID: W2052835726
description: scientific article; zbMATH DE number 1548155