Journal article

EXTINCTION IN LOWER HESSENBERG BRANCHING PROCESSES WITH COUNTABLY MANY TYPES

Peter Braunsteins, Sophie Hautphenne

Annals of Applied Probability | INST MATHEMATICAL STATISTICS | Published : 2019

Abstract

We consider a class of branching processes with countably many types which we refer to as Lower Hessenberg branching processes. These are multitype Galton–Watson processes with typeset X={0,1,2,…}, in which individuals of type i may give birth to offspring of type j≤i+1 only. For this class of processes, we study the set S of fixed points of the progeny generating function. In particular, we highlight the existence of a continuum of fixed points whose minimum is the global extinction probability vector q and whose maximum is the partial extinction probability vector q~. In the case where q~=1, we derive a global extinction criterion which holds under second moment conditions, and when q~<1 w..

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Grants

Awarded by Australian Research Council (ARC) through the Centre of Excellence for the Mathematical and Statistical Frontiers (ACEMS)


Funding Acknowledgements

Supported by the Australian Research Council (ARC) through the Centre of Excellence for the Mathematical and Statistical Frontiers (ACEMS) and the Discovery Project DP150101459.Supported by the Discovery Early Career Researcher Award DE150101044.