Coalescence times for the continuous time Bienaym\'e-Galton-Watson process

TL;DR

This paper studies the distribution of the coalescence time for two individuals in a continuous time Bienaymé-Galton-Watson process, providing limiting distributions and extensions to multiple individuals.

Contribution

It derives the distribution of coalescence times and their limits in the subcritical case, extending to joint distributions for multiple individuals.

Findings

Distribution of coalescence times characterized

Limiting distributions obtained for subcritical case

Extension to joint distributions for multiple individuals

Abstract

We investigate the distribution of the coalescence time (most recent common ancestor) for two individuals picked at random (uniformly) in the current generation of a continuous time Bienaym\'e-Galton-Watson process founded $t$ units of time ago. We also obtain limiting distributions as $t\rightarrow \infty $ in the subcritical case. We may also extend our results for two individuals to the joint distribution of coalescence times for any finite number of individuals sampled in the current generation.