arXiv Open Access 2026

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

Oleksii Galganov Andrii Ilienko

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Abstrak

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.

Topik & Kata Kunci

math.PR

Penulis (2)

Oleksii Galganov

Andrii Ilienko

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APA MLA BibTeX

Galganov, O., Ilienko, A. (2026). Explosion and non-explosion in pure birth Crump--Mode--Jagers branching processes. https://arxiv.org/abs/2601.06850

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Informasi Jurnal

Tahun Terbit: 2026
Bahasa: en
Sumber Database: arXiv
Akses: Open Access ✓