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Extinction in lower Hessenberg branching processes with countably many types. (English) Zbl 1439.60079
Summary: 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 \(\boldsymbol{\tilde{q}} \).

60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60J05 Discrete-time Markov processes on general state spaces
60J22 Computational methods in Markov chains
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