Invariant probability measures for a class of Feller Markov chains
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.
scientific article; zbMATH DE number 1548155