Approximation of the invariant probability measure of an infinite stochastic matrix

Abstract

Let P denote an irreducible positive recurrent infinite stochastic matrix with the unique invariant probability measure π. We consider sequences {P m }m∊N of stochastic matrices converging to P (pointwise), such that every Pm has at least one invariant probability measure π m . The aim of this paper is to find conditions, which assure that at least one of sequences {π m }m∊N converges to π (pointwise). This includes the case where the P m are finite matrices, which is of special interest. It is shown that there is a sequence of finite stochastic matrices, which can easily be constructed, such that {π m }m∊N converges to π. The conditions given for the general case are closely related to Foster's condition.