Explosion and non-explosion in pure birth Crump--Mode--Jagers branching processes
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915720904835072 |
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| author | Galganov, Oleksii Ilienko, Andrii |
| author_facet | Galganov, Oleksii Ilienko, Andrii |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_06850 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Explosion and non-explosion in pure birth Crump--Mode--Jagers branching processes Galganov, Oleksii Ilienko, Andrii Probability 60J80, 05C80 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. |
| title | Explosion and non-explosion in pure birth Crump--Mode--Jagers branching processes |
| topic | Probability 60J80, 05C80 |
| url | https://arxiv.org/abs/2601.06850 |