Reduced critical processes for small populations

TL;DR

This paper investigates the structure of critical Galton-Watson processes with small populations, focusing on the behavior of particles with descendants at a fixed future time under certain population constraints.

Contribution

It introduces a framework for analyzing the structure of critical branching processes with bounded populations, extending understanding of their long-term behavior.

Findings

Characterization of process structure under population constraints

Asymptotic behavior for different growth functions of ta(n)

Insights into the genealogy of particles with descendants at a fixed time

Abstract

Let $\left\{ Z(n),n\geq 1\right\} $ be a critical Galton-Watson branching process with finite variance for the offspring size of particles. Assuming that $0<Z(n)\leq \varphi (n)$, where either $\varphi (n)=an$ for some $a>0$ or $\varphi (n)=o(n)$ as $n\rightarrow \infty $, we study the structure of the process $% \left\{ Z(m,n),0\leq m\leq n\right\} ,$ where $Z(m,n)$ is the number of particles in the process at moment $m\leq n$ having a positive number of descendants at moment $n$.