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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 

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