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

We study the total branch length LnLn of the Bolthausen–Sznitman coalescent as the sample size nn tends to infinity. Asymptotic expansions for the moments of LnLn are presented. It is shown that Ln/E(Ln) converges to 1 in probability and that LnLn, properly normalized, converges weakly to a stable random variable as nn 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>0r>0. Moreover, the results show that, for the Bolthausen–Sznitman coalescent, the total branch length LnLn is closely related to XnXn, 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.