Asymptotic results concerning the total branch length of the Bolthausen–Sznitman coalescent

Abstract

We study the total branch length L n of the Bolthausen–Sznitman coalescent as the sample size n tends to infinity. Asymptotic expansions for the moments of L n are presented. It is shown that L n / E ( L n ) converges to 1 in probability and that L n , properly normalized, converges weakly to a stable random variable as n tends to infinity. The results are applied to derive a corresponding limiting law for the total number of mutations for the Bolthausen–Sznitman coalescent with mutation rate r > 0 . Moreover, the results show that, for the Bolthausen–Sznitman coalescent, the total branch length L n is closely related to X n , the number of collision events that take place until there is just a single block. The proofs are mainly based on an analysis of random recursive equations using associated generating functions.