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September 2012 Smaller population size at the MRCA time for stationary branching processes
Yu-Ting Chen, Jean-François Delmas
Ann. Probab. 40(5): 2034-2068 (September 2012). DOI: 10.1214/11-AOP668

Abstract

We consider an elementary model of random size varying population governed by a stationary continuous-state branching process. We compute the distributions of various variables related to the most recent common ancestor (MRCA): the time to the MRCA, the size of the current population and the size of the population just before the MRCA. In particular we observe a natural mild bottleneck effect as the size of the population just before the MRCA is stochastically smaller than the size of the current population. We also compute the number of individuals involved in the last coalescent event of the genealogical tree, that is, the number of individuals at the time of the MRCA who have descendants in the current population. By studying more precisely the genealogical structure of the population, we get asymptotics for the number of ancestors just before the current time. We give explicit computations in the case of the quadratic branching mechanism. In this case, the size of the population at the MRCA is, in mean, $2/3$ of the size of the current population. We also provide in this case the fluctuations for the renormalized number of ancestors.

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Yu-Ting Chen. Jean-François Delmas. "Smaller population size at the MRCA time for stationary branching processes." Ann. Probab. 40 (5) 2034 - 2068, September 2012. https://doi.org/10.1214/11-AOP668

Information

Published: September 2012
First available in Project Euclid: 8 October 2012

zbMATH: 1275.92076
MathSciNet: MR3025710
Digital Object Identifier: 10.1214/11-AOP668

Subjects:
Primary: 60J80 , 60J85 , 92D25
Secondary: 60G10 , 60G55 , 60J60

Keywords: bottleneck , branching process , Feller diffusion , genealogy , last coalescent event , Lévy tree , most recent common ancestor , random size population

Rights: Copyright © 2012 Institute of Mathematical Statistics

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Vol.40 • No. 5 • September 2012
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