Critical Galton-Watson processes with overlapping generations

TL;DR

This paper extends classical results on critical Galton-Watson processes by incorporating overlapping generations, establishing convergence of multiple population counts to a new class of limiting distributions represented through integrals involving the limiting process.

Contribution

It introduces a generalized framework for critical Galton-Watson processes with overlapping generations and characterizes the convergence of finite-dimensional distributions.

Findings

Convergence of scaled population counts to integral-based distributions.

Representation of limiting distributions via integrals of the limiting process.

Extension of classical Galton-Watson results to overlapping generations.

Abstract

A properly scaled critical Galton-Watson process converges to a continuous state critical branching process $\xi(\cdot)$ as the number of initial individuals tends to infinity. We extend this classical result by allowing for overlapping generations and considering a wide class of population counts. The main result of the paper establishes a convergence of the finite dimensional distributions for a scaled vector of multiple population counts. The set of the limiting distributions is conveniently represented in terms of integrals $(\int_0^y\xi(y-u)du^\gamma, y\ge0)$ with a pertinent $\gamma\ge0$.