Geometric ergodicity in a class of denumerable Markov chains

We study the question of geometric ergodicity in a class of Markov chains on the state space of non-negative integers for which, apart from a finite number of boundary rows and columns, the elements pjk of the one-step transition matrix are of the form c _k-j where {c _k} is a probability distribution on the set of integers. Such a process may be described as a general random walk on the non-negative integers with boundary conditions affecting transition probabilities into and out of a finite set of boundary states. The imbedded Markov chains of several non-Markovian queueing processes are special cases of this form. It is shown that there is an intimate connection between geometric ergodicity and geometric bounds on one of the tails of the distribution {c _k}.This research was supported by the U.S. office of Naval Research Contract No. Nonr-855(09), and carried out while the author was a visitor in the Statistics department, University of North Carolina, Chapel Hill.