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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 $$\boldsymbol{\tilde{q}}$$.

MSC:
 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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