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Quasi-limiting distributions of Markov chains that are skip-free to the left in continuous time

Part of: Markov processes Probability theory on algebraic and topological structures

Published online by Cambridge University Press:  14 July 2016

Masaaki Kijima
Masaaki Kijima*
Affiliation:
University of Tsukuba, Tokyo
*
∗ Postal address: Graduate School of Systems Management, University of Tsukuba, Tokyo, 3-29-1 Otsuka, Bunkyo-ku, Tokyo 112, Japan.
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Abstract

A continuous-time Markov chain on the non-negative integers is called skip-free to the left (right) if the governing infinitesimal generator A = (aij) has the property that aij = 0 for j ≦ i ‒ 2 (i ≦ j – 2). If a Markov chain is skip-free both to the left and to the right, it is called a birth-death process. Quasi-limiting distributions of birth–death processes have been studied in detail in their own right and from the standpoint of finite approximations. In this paper, we generalize, to some extent, results for birth-death processes to Markov chains that are skip-free to the left in continuous time. In particular the decay parameter of skip-free Markov chains is shown to have a similar representation to the birth-death case and a result on convergence of finite quasi-limiting distributions is obtained.

Keywords

QUASI-STATIONARY DISTRIBUTIONSKIP-FREE MARKOV CHAINBIRTH-DEATH PROCESSINVARIANT MEASUREPOLYNOMIAL

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60B10: Convergence of probability measures
Type
Research Papers
Information
Journal of Applied Probability , Volume 30 , Issue 3 , September 1993 , pp. 509 - 517
Copyright
Copyright © Applied Probability Trust 1993 

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