Ratio ergodicity for non-homogeneous Markov Chains in general state spaces

Abstract

In Refs. 7, 8, 12–14, applications of Birkhoff's coefficient of ergodicity to the study of products of finite and infinite matrices are given. In this paper we intend to show that using the ideas and results of Birkhoff and Baueret al. one can establish such results in the general setting of positive operators on a space of measures and in particular, in the study of nonhomogeneous Markov Chains in general state spaces. In this setup the proofs are very simple. Since the high level of generality in Ref. 1 hinders an easy understanding of the special case of interest to probabilists, we present a self-contained treatment. Even though our aim is methodological, our theorems are new, even in the case of infinite matrices.