On Existence of Limiting Distribution for Time-Nonhomogeneous Countable Markov Process

In this paper, sufficient conditions are given for the existence of limiting distribution of a nonhomogeneous countable Markov chain with time-dependent transition intensity matrix. The method of proof exploits the fact that if the distribution of random process Q=(Q _t ) _t≥0 is absolutely continuous with respect to the distribution of ergodic random process Q°=(Q° _t ) _t≥0, then $Q_t \xrightarrow{t \to \infty }]{{law}}\pi $ where π is the invariant measure of Q°. We apply this result for asymptotic analysis, as t→∞, of a nonhomogeneous countable Markov chain which shares limiting distribution with an ergodic birth-and-death process.