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

  • Authors:

  • V. Abramov;R. Liptser

  • Affiliations:

  • Department of Mathematics, The Faculty of Exact Sciences, Tel Aviv University, 69978 Tel Aviv, and College of Judea and Samaria, 44837 Ariel, Israel vyachesl@zahav.net.il;Department of Electrical Engineering-Systems, Tel Aviv University, 69978 Tel Aviv, Israel liptser@eng.tau.ac.il

  • Venue:
  • Queueing Systems: Theory and Applications
  • Year:
  • 2004

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Abstract

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 \[Q_{t}\mathop{\longrightarrow}\limits_{t\to\infty}^{\mathrm{law}}\pi,\] 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.