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Minimal Clade Size in the Bolthausen-Sznitman Coalescent

Part of: Combinatorial probability Stochastic processes Graph theory Markov processes Limit theorems

Published online by Cambridge University Press:  30 January 2018

Fabian Freund  and
Arno Siri-Jégousse
Fabian Freund*
Affiliation:
University of Hohenheim
Arno Siri-Jégousse*
Affiliation:
Centro de Investigación en Matemáticas, A.C.
*
∗ Postal address: Crop Plant Biodiversity and Breeding Informatics Group (350b), Institute of Plant Breeding, Seed Science and Population Genetics, University of Hohenheim, Fruwirthstrasse 21, 70599 Stuttgart, Germany. Email address: fabian.freund@uni-hohenheim.de
∗∗ Postal address: Centro de Investigación en Matemáticas, Calle Jalisco s/n, Col. Mineral de Valenciana, 36240 Guanajuato, Mexico. Email address: arno@cimat.mx
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Abstract

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In this article we show the asymptotics of distribution and moments of the size Xn of the minimal clade of a randomly chosen individual in a Bolthausen-Sznitman n-coalescent for n → ∞. The Bolthausen-Sznitman n-coalescent is a Markov process taking states in the set of partitions of {1, …, n}, where 1, …, n are referred to as individuals. The minimal clade of an individual is the equivalence class the individual is in at the time of the first coalescence event this individual participates in. We also provide exact formulae for the distribution of Xn. The main tool used is the connection of the Bolthausen-Sznitman n-coalescent with random recursive trees introduced by Goldschmidt and Martin (2005). With it, we show that Xn - 1 is distributed as the size of a uniformly chosen table in a standard Chinese restaurant process with n - 1 customers.

Keywords

Minimal clade sizeBolthausen-Sznitman n-coalescentChinese restaurant process

MSC classification

Primary: 60C05: Combinatorial probability
Secondary: 05C80: Random graphs 60G09: Exchangeability 60F05: Central limit and other weak theorems 60J27: Continuous-time Markov processes on discrete state spaces 92D25: Population dynamics (general)
Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 3 , September 2014 , pp. 657 - 668
Copyright
© Applied Probability Trust 

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