Convergence of nonhomogeneous stochastic chains with countable states

The work started by [4], Theory Probab. Appl. 15, 604-618], and continued by [6], Trans. Amer. Math. Soc. 263, 505-520], is extended, and completed with respect to certain aspects. Infinite-dimensional stochastic chains are considered in the framework of Mukherjea [loc. cit.]; backward products of stochastic matrices and their convergence are also considered. The main theme centers around understanding how the convergence of products (backward and forward, finite and infinite dimensional) takes place and what it means in terms of various types of asymptotic behavior of the individual stochastic matrices in the chain. The study is based on establishing the existence of a basis for convergent chains. The basis then makes it possible to describe properly various aspects of convergence. All results are new; they are also complete at least in the sense they have been presented and suitable examples (or counter-examples) are presented to justify the assumptions involved.