Necessary and sufficient conditions for the ergodicity of Markov chains with transition Δm,n(Δ′m,n)-matrix

Abstract

This paper isolates and studies a class of Markov chains with a special quasi-triangular form of the transition matrix [so-called Δm,n(Δ′m,n)-matrix]. Many discrete stochastic processes encountered in applications (queues, inventories and dams) have transition matrices which are special cases of a Δm,n(Δ′m,n)-matrix. Necessary and sufficient conditions for the ergodicity of a Markov chain with transition Δm,n(Δ′m,n)-matrix are determined in the article in two equivalent versions. According to the first version, these conditions are expressed in terms of certain restrictions imposed on the generating functions Ai(x) of the elements of the i-th row of the transition matrix, i=0,1,2,…; in the other version they are connected with the characterization of the roots of a certain associated function in the unit circle of the complex plane. Results obtained in the article generalize, complement, and refine similar results existing in the literature.