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Total Variation Approximation for Quasi-Stationary Distributions

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

A. D. Barbour  and
P. K. Pollett
A. D. Barbour*
Affiliation:
Universität Zürich
P. K. Pollett*
Affiliation:
University of Queensland
*
∗Postal address: Angewandte Mathematik, Universität Zürich, Winterthurertrasse 190, CH-8057 Zürich, Switzerland. Email address: a.d.barbour@math.uzh.ch
∗∗Postal address: Department of Mathematics, University of Queensland, Brisbane, QLD 4072, Australia.
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Abstract

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Quasi-stationary distributions, as discussed in Darroch and Seneta (1965), have been used in biology to describe the steady state behaviour of population models which, while eventually certain to become extinct, nevertheless maintain an apparent stochastic equilibrium for long periods. These distributions have some drawbacks: they need not exist, nor be unique, and their calculation can present problems. In this paper, we give biologically plausible conditions under which the quasi-stationary distribution is unique, and can be closely approximated by distributions that are simple to compute.

Keywords

Quasi-stationary distributiontotal variation distancestochastic logistic model

MSC classification

Secondary: 60J28: Applications of continuous-time Markov processes on discrete state spaces 92D25: Population dynamics (general) 92D30: Epidemiology
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 4 , December 2010 , pp. 934 - 946
Copyright
Copyright © Applied Probability Trust 2010 

Footnotes

Research supported in part by Schweizerischer Nationalfonds Projekt Nr. 20-107935/1.

Research supported in part by the Australian Research Council Centre of Excellence for Mathematics and Statistics of Complex Systems.

References

Anderson, W. J. (1991). Continuous-Time Markov Chains. Springer, New York.CrossRefGoogle Scholar
Bartlett, M. S. (1960). Stochastic Population Models in Ecology and Epidemiology. Methuen, London.Google Scholar
Cattiaux, P. et al. (2009). Quasi-stationary distributions and diffusion models in population dynamics. Ann. Prob. 5, 1926–1926.Google Scholar
Darroch, J. N. and Seneta, E. (1965). On quasi-stationary distributions in absorbing discrete-time Markov chains. J. Appl. Prob. 2, 88–100.Google Scholar
Ewens, W. J. (1963). The diffusion equation and a pseudo-distribution in genetics. J. R. Statist. Soc. B 25, 405–412.Google Scholar
Ewens, W. J. (1964). The pseudo-transient distribution and its uses in genetics. J. Appl. Prob. 1, 141–156.CrossRefGoogle Scholar
Ferrari, P. A., Kesten, H., Martínez, S. and Picco, P. (1995). Existence of quasi-stationary distributions. A renewal dynamic approach. Ann. Prob. 23, 501–521.Google Scholar
Lindvall, T. (2002). Lectures on the Coupling Method, 2nd edn. Dover, Mineola, NY.Google Scholar
Pitman, J. W. (1974). Uniform rates of convergence for Markov chain transition probabilities. Z. Wahrscheinlichkeitsth. 29, 193–227.Google Scholar
Steinsaltz, D. and Evans, S. N. (2004). Markov mortality models: implications of quasistationarity and varying initial distributions. Theoret. Pop. Biol. 65, 319–337.Google Scholar
Van Doorn, E. A. (1991). Quasi-stationary distributions and convergence to quasi-stationarity of birth-death processes. Adv. Appl. Prob. 23, 683–700.Google Scholar
Verhulst, P.-F. (1838). Notice sur la loi que la population poursuit dans son accroissement. Corresp. Math. Phys. 10, 113–121.Google Scholar
Yaglom, A. M. (1947). Certain limit theorems of the theory of branching processes. Dok. Akad. Nauk SSSR 56, 795–798.Google Scholar