Critical branching as a pure death process coming down from infinity

TL;DR

This paper studies a critical Galton-Watson process with overlapping generations, showing it converges to a pure death process from infinity, offering new insights into population dynamics in critical regimes.

Contribution

It establishes the convergence of the process conditioned on non-extinction to a pure death process, providing a new perspective on age-dependent population behavior.

Findings

Convergence of finite-dimensional distributions conditioned on non-extinction.

Identification of the limiting process as a pure death process from infinity.

Insight into Vatutin's dichotomy in critical age-dependent reproduction.

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.