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The probabilities of extinction in a branching random walk on a strip

Part of: Markov processes

Published online by Cambridge University Press:  04 September 2020

Peter Braunsteins  and
Sophie Hautphenne
Peter Braunsteins*
Affiliation:
The University of Melbourne
Sophie Hautphenne*
Affiliation:
The University of Melbourne
*
*Postal address: School of Mathematics and Statistics, The University of Melbourne, 3010, Melbourne, Australia.
*Postal address: School of Mathematics and Statistics, The University of Melbourne, 3010, Melbourne, Australia.
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Abstract

We consider a class of multitype Galton–Watson branching processes with a countably infinite type set $\mathcal{X}_d$ whose mean progeny matrices have a block lower Hessenberg form. For these processes, we study the probabilities $\textbf{\textit{q}}(A)$ of extinction in sets of types $A\subseteq \mathcal{X}_d$ . We compare $\textbf{\textit{q}}(A)$ with the global extinction probability $\textbf{\textit{q}} = \textbf{\textit{q}}(\mathcal{X}_d)$ , that is, the probability that the population eventually becomes empty, and with the partial extinction probability $\tilde{\textbf{\textit{q}}}$ , that is, the probability that all types eventually disappear from the population. After deriving partial and global extinction criteria, we develop conditions for $\textbf{\textit{q}} < \textbf{\textit{q}}(A) < \tilde{\textbf{\textit{q}}}$ . We then present an iterative method to compute the vector $\textbf{\textit{q}}(A)$ for any set A. Finally, we investigate the location of the vectors $\textbf{\textit{q}}(A)$ in the set of fixed points of the progeny generating vector.

Keywords

Infinite-type branching processextinction probabilityfixed point

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J05: Discrete-time Markov processes on general state spaces 60J22: Computational methods in Markov chains
Type
Research Papers
Information
Journal of Applied Probability , Volume 57 , Issue 3 , September 2020 , pp. 811 - 831
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Copyright
© Applied Probability Trust 2020

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