Journal of Applied Probability

Minimal clade size in the Bolthausen-Sznitman coalescent

Fabian Freund and Arno Siri-Jégousse

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Abstract

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.

Article information

Source
J. Appl. Probab. Volume 51, Number 3 (2014), 657-668.

Dates
First available in Project Euclid: 5 September 2014

Permanent link to this document
http://projecteuclid.org/euclid.jap/1409932665

Digital Object Identifier
doi:10.1239/jap/1409932665

Mathematical Reviews number (MathSciNet)
MR3256218

Zentralblatt MATH identifier
1320.60132

Subjects
Primary: 60C05: Combinatorial probability
Secondary: 05C80: Random graphs [See also 60B20] 60G09: Exchangeability 60F05: Central limit and other weak theorems 60J27: Continuous-time Markov processes on discrete state spaces 92D25: Population dynamics (general)

Keywords
Minimal clade size Bolthausen-Sznitman n-coalescent Chinese restaurant process

Citation

Freund, Fabian; Siri-Jégousse, Arno. Minimal clade size in the Bolthausen-Sznitman coalescent. J. Appl. Probab. 51 (2014), no. 3, 657--668. doi:10.1239/jap/1409932665. http://projecteuclid.org/euclid.jap/1409932665.


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