Explosion and non-explosion in pure birth Crump--Mode--Jagers branching processes

Oleksii Galganov; Andrii Ilienko

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arXiv · 2601.06850

Explosion and non-explosion in pure birth Crump--Mode--Jagers branching processes

Oleksii Galganov·Andrii Ilienko

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Abstract

In this short note, we provide an explicit sufficient condition for non-explosion of Crump--Mode--Jagers branching processes with pure birth reproduction. It shows that the standard sufficient condition for explosion, namely the convergence of the series of reciprocals of the birth rates, is -- at least for rate sequences without excessive oscillations -- remarkably close to being necessary. At the same time, it is not necessary in full generality: we construct a counterexample which also yields a general preferential attachment tree without fitness with an infinite path and no vertices of infinite degree, thereby answering an open question previously raised in the literature.

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Oleksii Galganov, Andrii Ilienko. 2026-01-11. Explosion and non-explosion in pure birth Crump--Mode--Jagers branching processes. https://arxiv.org/abs/2601.06850

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