Coalescence times for critical Galton-Watson processes with immigration

TL;DR

This paper investigates the asymptotic behavior of coalescence times in critical Galton-Watson processes with immigration, providing insights into the genealogical structure of such stochastic populations.

Contribution

It introduces new results on the limiting distributions of coalescence times for particles and the entire population in critical Galton-Watson processes with immigration.

Findings

Limiting behavior of pairwise coalescence times characterized

Asymptotic distribution of population coalescence time derived

Insights into genealogical structures of critical processes obtained

Abstract

Let $X^I_n$ be the coalescence time of two particles picked at random from the $n$th generation of a critical Galton-Watson process with immigration, and let $A^I_n$ be the coalescence time of the whole population in the $n$th generation. In this paper, we study the limiting behaviors of $X^I_n$ and $A^I_n$ as $n\to\infty$.