Critical branching as a pure death process coming down from infinity

Serik Sagitov

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arXiv · 2111.04274

Critical branching as a pure death process coming down from infinity

Serik Sagitov

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Abstract

We consider the critical Galton-Watson process with overlapping generations stemming from a single founder. Assuming that both the variance of the offspring number and the average generation length are finite, we establish the convergence of the finite-dimensional distributions, conditioned on non-extinction at a remote time of observation. The limiting process is identified as a pure death process coming down from infinity. This result brings a new perspective on Vatutin's dichotomy claiming that in the critical regime of age-dependent reproduction, an extant population either contains a large number of short-living individuals or consists of few long-living individuals.

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Serik Sagitov. 2021-11-08. Critical branching as a pure death process coming down from infinity. https://arxiv.org/abs/2111.04274

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