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Extinction Probabilities of Supercritical Decomposable Branching Processes

Part of: Markov processes

Published online by Cambridge University Press:  04 February 2016

Sophie Hautphenne
Sophie Hautphenne*
Affiliation:
Université Libre de Bruxelles and The University of Melbourne
*
∗ Postal address: Department of Mathematics and Statistics, The University of Melbourne, Parkville, VIC 3010, Australia. Email address: sophiemh@unimelb.edu.au
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Abstract

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We focus on supercritical decomposable (reducible) multitype branching processes. Types are partitioned into irreducible equivalence classes. In this context, extinction of some classes is possible without the whole process becoming extinct. We derive criteria for the almost-sure extinction of the whole process, as well as of a specific class, conditionally given the class of the initial particle. We give sufficient conditions under which the extinction of a class implies the extinction of another class or of the whole process. Finally, we show that the extinction probability of a specific class is the minimal nonnegative solution of the usual extinction equation but with added constraints.

Keywords

Multitype branching processdecomposable branching processextinction criteriaextinction probability

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Research Article
Information
Journal of Applied Probability , Volume 49 , Issue 3 , September 2012 , pp. 639 - 651
Copyright
© Applied Probability Trust 

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