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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.)

Information

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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References

Anderson, W. J. (1991) Continuous-Time Markov Chains. Springer-Verlag, New York.Google Scholar
Asmussen, S. and Hering, H. (1983) Branching Processes. Birkhauser, Boston.Google Scholar
Athreya, K. B. and Ney, P. E. (1972) Branching Processes. Springer-Verlag, Berlin.CrossRefGoogle Scholar
Brockwell, P. J., Gani, J. and Resnick, S. I. (1982) Birth, immigration and catastrophe processes. Adv. Appl. Prob. 14, 709–731.CrossRefGoogle Scholar
Cavender, J. A. (1978) Quasi-stationary distributions of birth-and-death processes. Adv. Appl. Prob. 10, 570–586.Google Scholar
Doorn, E. A. Van (1991) Quasi-stationary distributions and convergence to quasi-stationarity of birth–death processes. Adv. Appl. Prob. 23, 683–700.CrossRefGoogle Scholar
Eshel, I. (1981) On the survival probability of a slightly advantageous mutant gene with a general distribution of progeny size—A branching process model. J. Math. Biol. 12, 355–362.Google Scholar
Ferrari, P. A., Kesten, H., Martinez, S. and Picco, P. (1994) Existence of quasi-stationary distributions. A renewal dynamical approach. Ann. Prob. To appear.Google Scholar
Holgate, P. (1967) Divergent population processes and mammal outbreaks. J. Appl. Prob. 4, 1–8.Google Scholar
Kesten, H. (1993) A ratio limit theorem for (sub) Markov chains on {1, 2, …} with bounded jumps. Technical Report, Dept. Math., Cornell University.Google Scholar
Pakes, A. G. (1975) Conditional limit theorems for a left continuous random walk. J. Appl. Prob. 10, 39–53.Google Scholar
Pakes, A. G. (1978) The characterization of series amenable to ratio tests. Math. Acad. Serbe Sci. 23 (37), 155–162.Google Scholar
Pakes, A. G. (1979a) The age of a Markov process. Stoch. Proc. Appl. 8, 277–303.CrossRefGoogle Scholar
Pakes, A. G. (1979b) Limit theorems for the simple branching process allowing immigration, I: The case of finite offspring mean. Adv. Appl. Prob. 11, 31–62.Google Scholar
Pakes, A. G. (1986) The Markov branching-catastrophe process. Stoch. Proc. Appl. 23, 1–33.CrossRefGoogle Scholar
Pakes, A. G. (1989) Asymptotic results for the extinction time of Markov branching processes allowing emigration. I, Random walk decrements. Adv. Appl. Prob. 21, 243–269.CrossRefGoogle Scholar
Pakes, A. G. (1993) Population processes with hard culling. In preparation.Google Scholar
Pakes, A. G. and Pollett, P. (1989) The supercritical birth, death and catastrophe process: limit theorems on the set of extinction. Stoch. Proc. Appl. 32, 161–170.Google Scholar
Pollett, P. (1986) On the equivalence of µ-invariant measures for the minimal process and its q-matrix. Stoch. Proc. Appl. 22, 203–221.CrossRefGoogle Scholar
Raup, D. M. (1991) Extinction: Bad Genes or Bad Luck. W. W. Norton, New York.Google Scholar
Santana, P. S. (1987) A population-size dependent branching process. In New Perspectives in Theoretical and Applied Statistics , ed. Puri, M. L. et al. Wiley, New York.Google Scholar
Seneta, E. (1966) Quasi-stationary behaviour in the random walk with continuous time. Austral. J. Statist. 8, 92–98.Google Scholar
Vere-Jones, D. (1969) Some limit theorems for evanescent processes. Austral. J. Statist. 11, 67–78.Google Scholar