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

This paper isolates and studies a class of Markov chains with a special quasi-triangular form of the transition matrix [so-called Δ m , 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 Δ m , n ( Δ ′ m , n ) -matrix. Necessary and sufficient conditions for the ergodicity of a Markov chain with transition Δ m , 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 A i ( 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.