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May 2020 Branching random walks with uncountably many extinction probability vectors
Daniela Bertacchi, Fabio Zucca
Braz. J. Probab. Stat. 34(2): 426-438 (May 2020). DOI: 10.1214/19-BJPS430

Abstract

Given a branching random walk on a set $X$, we study its extinction probability vectors $\mathbf{q}(\cdot,A)$. Their components are the probability that the process goes extinct in a fixed $A\subseteq X$, when starting from a vertex $x\in X$. The set of extinction probability vectors (obtained letting $A$ vary among all subsets of $X$) is a subset of the set of the fixed points of the generating function of the branching random walk. In particular here we are interested in the cardinality of the set of extinction probability vectors. We prove results which allow to understand whether the probability of extinction in a set $A$ is different from the one of extinction in another set $B$. In many cases there are only two possible extinction probability vectors and so far, in more complicated examples, only a finite number of distinct extinction probability vectors had been explicitly found. Whether a branching random walk could have an infinite number of distinct extinction probability vectors was not known. We apply our results to construct examples of branching random walks with uncountably many distinct extinction probability vectors.

Citation

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Daniela Bertacchi. Fabio Zucca. "Branching random walks with uncountably many extinction probability vectors." Braz. J. Probab. Stat. 34 (2) 426 - 438, May 2020. https://doi.org/10.1214/19-BJPS430

Information

Received: 1 June 2018; Accepted: 1 January 2019; Published: May 2020
First available in Project Euclid: 4 May 2020

zbMATH: 07232937
MathSciNet: MR4093267
Digital Object Identifier: 10.1214/19-BJPS430

Rights: Copyright © 2020 Brazilian Statistical Association

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Vol.34 • No. 2 • May 2020
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