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 = (Qt )t 0 is absolutely continuous with respect to the distribution of ergodic random process Q◦ = (Q◦t )t 0, then Qt law −→ t→∞ 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-anddeath process.