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Markovian paths to extinction

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

Peter Jagers ,
Fima C. Klebaner  and
Serik Sagitov
Peter Jagers*
Affiliation:
Chalmers University of Technology
Fima C. Klebaner*
Affiliation:
Monash University
Serik Sagitov*
Affiliation:
Chalmers University of Technology
*
∗ Postal address: Department of Mathematical Sciences, Chalmers University of Technology, SE-412 96 Göteborg, Sweden.
∗∗∗ Postal address: School of Mathematical Sciences, Monash University, Clayton, VIC 3800, Australia.
∗ Postal address: Department of Mathematical Sciences, Chalmers University of Technology, SE-412 96 Göteborg, Sweden.
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Abstract

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Subcritical Markov branching processes {Zt} die out sooner or later, say at time T < ∞. We give results for the path to extinction {ZuT, 0 ≤ u ≤ 1} that include its finite dimensional distributions and the asymptotic behaviour of xu−1ZuT, as Z0=x → ∞. The limit reflects an interplay of branching and extreme value theory. Then we consider the population on the verge of extinction, as modelled by ZT-u, u > 0, and show that as Z0= x → ∞ this process converges to a Markov process {Yu}, which we describe completely. Emphasis is on continuous time processes, those in discrete time displaying a more complex behaviour, related to Martin boundary theory.

Keywords

Branching processMarkov processextinctionextreme value

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 92D25: Population dynamics (general)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 39 , Issue 2 , June 2007 , pp. 569 - 587
Copyright
Copyright © Applied Probability Trust 2007 

References

Alsmeyer, G. and Rösler, U. (2002). Asexual versus promiscuous bisexual Galton–Watson processes: the extinction probability ratio. Ann. Appl. Prob. 12, 125–142.Google Scholar
Alsmeyer, G. and Rösler, U. (2006). The Martin entrance boundary of the Galton–Watson process. Ann. Inst. H. Poincaré Prob. Statist. 42, 591–606.Google Scholar
Athreya, K. and Ney, P. (1972). Branching Processes. Springer, Berlin.CrossRefGoogle Scholar
Billingsley, P. (1999). Convergence of Probability Measures, 2nd edn. John Wiley, New York.CrossRefGoogle Scholar
Gut, A. and Janson, S. (2001). Tightness and weak convergence for Jump processes. Statist. Prob. Lett. 52, 101–107.CrossRefGoogle Scholar
Harris, T. E. (1963). The Theory of Branching Processes. Springer, Berlin.CrossRefGoogle Scholar
Jagers, P. (1975). Branching Processes with Biological Applications. John Wiley, Chichester.Google Scholar
Jagers, P., Klebaner, F. C. and Sagitov, S. (2007). On the path to extinction. Proc. Nat. Acad. Sci. USA 104, 6107–6111.Google Scholar
Klebaner, F. C., Rösler, U. and Sagitov, S. (2006). Transformations of Galton–Watson processes and linear fractional reproduction. Submitted.Google 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. (1989). On the asymptotic behaviour of the extinction time of the simple branching process. Adv. Appl. Prob. 21, 470–472.Google Scholar
Sevast'yanov, B. A. (1971). Vetvyashchiesya Protsessy. Nauka, Moscow.Google Scholar