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Coalescence times for the branching process

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

Amaury Lambert
Amaury Lambert*
Affiliation:
Ecole Normale Supérieure, Paris
*
∗ Postal address: Unit of Mathematical Evolutionary Biology, Fonctionnement et Evolution des Systèmes Ecologiques UMR 7625, Ecole Normale Supérieure, 46, rue d'Ulm, F-75230 Paris Cedex 05, France. Email address: alambert@ens.fr
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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 founded t units of time ago, in both the discrete and continuous (time and state-space) settings. We obtain limiting distributions as t→∞ 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.

Keywords

GenealogyBienaymé-Galton-Watson processcontinuous-state branching processcoalescencequasi-stationary distribution

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 35 , Issue 4 , December 2003 , pp. 1071 - 1089
Copyright
Copyright © Applied Probability Trust 2003 

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