Minimal clade size in the Bolthausen-Sznitman coalescent

Abstract

This article shows 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 $\left\{1,\ldots,n\right\}$, where $1,\ldots,n$ 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.\\ The main tool used is the connection of the Bolthausen-Sznitman $n$-coalescent with random recursive trees introduced by Goldschmidt and Martin (see \cite{goldschmidtmartin}). This connection shows that $X_n-1$ is distributed as the number $M_n$ of all individuals not in the equivalence class of individual 1 shortly before the time of the last coalescence event. Both functionals are distributed like the size $RT_{n-1}$ of an uniformly chosen table in a standard Chinese restaurant process with $n-1$ customers.We give exact formulae for these distributions.\\ Using the asymptotics of $M_n$ shown by Goldschmidt and Martin in \cite{goldschmidtmartin}, we see $(\log n)^{-1}\log X_n$ converges in distribution to the uniform distribution on [0,1] for $n\to\infty$.\\ We provide the complimentary information that $\frac{\log n}{n^k}E(X_n^k)\to \frac{1}{k}$ for $n\to\infty$, which is also true for $M_n$ and $RT_n$.