Ergodic theorems for sequences of infinite stochastic matrices

Abstract

In the theory of finite state, discrete time, non-homogeneous Markov chains, different notions of ergodicity have been introduced in the literature. These notions are concerned with the long-run behaviour of chains and with their tendency to get some stability properties after a sufficiently long period of time. The aim of this paper is the study of non-homogeneous Markov chains with a denumerable number of states. It will be shown that some theorems which are valid in the finite case are also valid for chains with a denumerable number of states as well. Moreover, a new notion of stability is introduced and it is shown to be satisfied for some chains. Although the paper is self-contained some familiarity with the theory of finite state non-homogeneous Markov chains is desired. Without any attempt of completeness we list for the interested reader the papers: (1–3, 8, 9).