On the genealogy and coalescence times of Bienaym\'e-Galton-Watson branching processes

TL;DR

This paper analyzes the ancestral genealogy and coalescence times of Bienaymé-Galton-Watson branching processes, providing new analytical tools and explicit formulas for coalescence times and genealogy structure.

Contribution

It introduces an analytical framework for studying genealogy in unconditional Bienaymé-Galton-Watson processes, including transition matrices and integral representations of coalescence times.

Findings

Derived the transition matrix of the ancestral count process.

Provided integral formulas for coalescence time distributions.

Obtained closed-form probability generating functions for key examples.

Abstract

Coalescence processes have received a lot of attention in the context of conditional branching processes with fixed population size and non-overlapping generations. Here we focus on similar problems in the context of the standard unconditional Bienaym\'e-Galton-Watson branching processes, either (sub)-critical or supercritical. Using an analytical tool, we derive the structure of some counting aspects of the ancestral genealogy of such processes, including: the transition matrix of the ancestral count process and an integral representation of various coalescence times distributions, such as the time to most recent common ancestor of a random sample of arbitrary size, including full size. We illustrate our results on two important examples of branching mechanisms displaying either finite or infinite reproduction mean, their main interest being to offer a closed form expression for their probability generating functions at all times. Large time behaviors are investigated.