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Quasi-stationary laws for Markov processes: examples of an always proximate absorbing state

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

Anthony G. Pakes
Anthony G. Pakes*
Affiliation:
University of Western Australia
*
* Postal address: Department of Mathematics, University of Western Australia, Nedlands, WA 6009, Australia.
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Abstract

Under consideration is a continuous-time Markov process with non-negative integer state space and a single absorbing state 0. Let T be the hitting time of zero and suppose Pi(T < ∞) ≡ 1 and (*) limi→∞Pi(T > t) = 1 for all t > 0. Most known cases satisfy (*). The Markov process has a quasi-stationary distribution iff Ei (e∊T) < ∞ for some ∊ > 0.

The published proof of this fact makes crucial use of (*). By means of examples it is shown that (*) can be violated in quite drastic ways without destroying the existence of a quasi-stationary distribution.

Keywords

INVARIANT MEASURESQUASI-STATIONARY DISTRIBUTIONLIMITING CONDITIONAL DISTRIBUTIONBIRTH PROCESSBRANCHING PROCESS

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 27 , Issue 1 , March 1995 , pp. 120 - 145
Copyright
Copyright © Applied Probability Trust 1995 

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