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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 $\mathcal{X}=\{0,1,2,\ldots\}$, in which individuals of type $i$ may give birth to offspring of type $j\leq i+1$ 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 $\boldsymbol{q}$ and whose maximum is the partial extinction probability vector $\boldsymbol{\tilde{q}}$. In the case where $\boldsymbol{\tilde{q}}=\boldsymbol{1}$, we derive a global extinction criterion which holds under second moment conditions, and when $\boldsymbol{\tilde{q}}<\boldsymbol{1}$ we develop necessary and sufficient conditions for $\boldsymbol{q}=\boldsymbol{\tilde{q}}$. We also correct a result in the literature on a sequence of finite extinction probability vectors that converge to the infinite vector $\boldsymbol{\tilde{q}}$.

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

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.29 • No. 5 • October 2019
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