Coalescence times for the branching process

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 branching process foundedtunits of time ago, in both the discrete and continuous (time and state-space) settings. We obtain limiting distributions ast→∞ in the subcritical case. In the continuous setting, these distributions are specified for quadratic branching mechanisms (corresponding to Brownian motion and Brownian motion with positive drift), and we 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.