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

AbstractWe study the total branch length Ln of the Bolthausen–Sznitman coalescent as the sample size n tends to infinity. Asymptotic expansions for the moments of Ln are presented. It is shown that Ln/E(Ln) converges to 1 in probability and that Ln, 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 Ln is closely related to Xn, 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