Minimal clade size in the Bolthausen-Sznitman coalescent. (English) Zbl 1320.60132

Summary: 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 \(n\to\infty\). The Bolthausen-Sznitman \(n\)-coalescent is a Markov process taking states in the set of partitions of \(\{1,\dots, n\}\), where \(1,\dots, n\) 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 C. Goldschmidt and J. B. Martin [Electron. J. Probab. 10, Paper No. 21, 718–745 (2005; Zbl 1109.60060)]. 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.


60J27 Continuous-time Markov processes on discrete state spaces
92D25 Population dynamics (general)
60C05 Combinatorial probability
05C80 Random graphs (graph-theoretic aspects)
60G09 Exchangeability for stochastic processes
60F05 Central limit and other weak theorems


Zbl 1109.60060
Full Text: DOI arXiv Euclid


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