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Minimal Clade Size in the Bolthausen-Sznitman Coalescent

Minimal Clade Size in the Bolthausen-Sznitman Coalescent In this article we show the asymptotics of distribution and moments of the size X n 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 X n . 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 X n - 1 is distributed as the size of a uniformly chosen table in a standard Chinese restaurant process with n - 1 customers. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Applied Probability Cambridge University Press

Minimal Clade Size in the Bolthausen-Sznitman Coalescent

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Publisher
Cambridge University Press
Copyright
© Applied Probability Trust 
ISSN
1475-6072
eISSN
0021-9002
DOI
10.1239/jap/1409932665
Publisher site
See Article on Publisher Site

Abstract

In this article we show the asymptotics of distribution and moments of the size X n 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 X n . 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 X n - 1 is distributed as the size of a uniformly chosen table in a standard Chinese restaurant process with n - 1 customers.

Journal

Journal of Applied ProbabilityCambridge University Press

Published: Jan 30, 2018

Keywords: Minimal clade size; Bolthausen-Sznitman n-coalescent; Chinese restaurant process; 60C05; 05C80; 60G09; 60F05; 60J27; 92D25

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