Minimal clade size in the Bolthausen-Sznitman coalescent

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.