Convergence of nonhomogeneous stochastic chains with countable states (Q2266508)

Let \((P_ n)\) be a sequence of finite or infinite-dimensional stochastic matrices. The authors study convergent (as \(n\to \infty)\) products of the forward type \(P_{k,n}=P_{k+1}P_{k+2}...P_ n\) as well as of the backward type \(P_{n,k}=P_ nP_{n-1}...P_{k+1}\). The main theme centers around understanding how the convergence of such products take place and what it means in terms of various types of asymptotic behaviour of the individual matrices in \((P_ n).\) The study is based on establishing the existence of a basis for convergent chains. A basis is a partition of the state space of the chain which for homogeneous products consists of the recurrent classes and the set of transient states. The main results are of the type \(\sum^{\infty}_{n=1}(P_ n)_{ij}<\infty\) for i,j in different classes of states in the basis, and generalizations of such results. Throughout the paper the authors point out both differences and similarities in the ''forward'' and ''backward'' cases. Several examples and counterexamples are given to justify the assumptions and illustrate various difficulties.