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Quasistationarity in continuous-time Markov chains where absorption is not certain

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

S. J. Darlington  and
P. K. Pollett
S. J. Darlington*
Affiliation:
University of Queensland
P. K. Pollett*
Affiliation:
University of Queensland
*
∗Postal address: Department of Mathematics, University of Queensland, Qld 4072, Australia
∗Postal address: Department of Mathematics, University of Queensland, Qld 4072, Australia
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Abstract

In a recent paper [4] it was shown that, for an absorbing Markov chain where absorption is not guaranteed, the state probabilities at time t conditional on non-absorption by t generally depend on t. Conditions were derived under which there can be no initial distribution such that the conditional state probabilities are stationary. The purpose of this note is to show that these conditions can be relaxed completely: we prove, once and for all, that there are no circumstances under which a quasistationary distribution can admit a stationary conditional interpretation.

Keywords

μ-invariant measuresduality

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60J35: Transition functions, generators and resolvents
Type
Short Communications
Information
Journal of Applied Probability , Volume 37 , Issue 2 , June 2000 , pp. 598 - 600
Copyright
Copyright © by the Applied Probability Trust 2000 

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References

Anderson, W. (1991). Continuous-Time Markov Chains: An Applications-Oriented Approach. Springer, New York.CrossRefGoogle Scholar
Flaspohler, D. (1974). Quasi-stationary distributions for absorbing continuous-time denumerable Markov chains. Ann. Inst. Statist. Math. 26, 351–356.Google Scholar
Nair, M., and Pollett, P. (1993). On the relationship between μ-invariant measures and quasistationary distributions for continuous-time Markov chains. Adv. Appl. Prob. 25, 82–102.Google Scholar
Pollett, P. (1999). Quasistationary distributions for continuous-time Markov chains when absorption is not certain. J. Appl. Prob. 36, 268–272.Google Scholar