This paper isolates and studies a class of Markov chains with a special quasi-triangular form of the transition matrix [so-called \(\Delta_{m,n}\) \((\Delta '_{m,n})\)-matrix]. Many discrete stochastic processes encountered in applications (queues, inventories and dams) have transition matrices which are special cases of a \(\Delta_{m,n}(\Delta '_{m,n})\)-matrix. Necessary and sufficient conditions for the ergodicity of a Markov chain with transition \(\Delta_{m,n}(\Delta '_{m,n})\)-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.