Open Access
October 2019 Extinction in lower Hessenberg branching processes with countably many types
Peter Braunsteins, Sophie Hautphenne
Ann. Appl. Probab. 29(5): 2782-2818 (October 2019). DOI: 10.1214/19-AAP1464

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 ji+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 we develop necessary and sufficient conditions for q=q~. We also correct a result in the literature on a sequence of finite extinction probability vectors that converge to the infinite vector q~.

Citation

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Peter Braunsteins. Sophie Hautphenne. "Extinction in lower Hessenberg branching processes with countably many types." Ann. Appl. Probab. 29 (5) 2782 - 2818, October 2019. https://doi.org/10.1214/19-AAP1464

Information

Received: 1 July 2017; Revised: 1 January 2019; Published: October 2019
First available in Project Euclid: 18 October 2019

zbMATH: 07155059
MathSciNet: MR4019875
Digital Object Identifier: 10.1214/19-AAP1464

Subjects:
Primary: 60J80
Secondary: 60J05 , 60J22

Keywords: extinction criterion , extinction probability , fixed point , Infinite-type branching process , varying environment

Rights: Copyright © 2019 Institute of Mathematical Statistics

Vol.29 • No. 5 • October 2019
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