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Critical branching as a pure death process coming down from infinity

Part of: Markov processes

Published online by Cambridge University Press:  18 January 2023

Serik Sagitov Open the ORCID record for Serik Sagitov [Opens in a new window]
Serik Sagitov*
Affiliation:
Chalmers University of Technology and University of Gothenburg
*
*Postal address: Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, SE-412 96 Göteborg, Sweden. Email address: serik@chalmers.se
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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.

Keywords

Galton–Watson process with overlapping generationsBellman–Harris processSevastyanov processCrump–Mode–Jagers processconvergence of finite-dimensional distributionsVatutin’s dichotomy

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Original Article
Information
Journal of Applied Probability , Volume 60 , Issue 2 , June 2023 , pp. 607 - 628
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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