Asymptotics of the allele frequency spectrum associated with the Bolthausen-Sznitman coalescent

We imagine a coalescent process as modelling the genealogy of a sample from a population which is subject to neutral mutation. We work under the assumptions of the infinitely many alleles model so that, in particular, every mutation gives rise to a completely new type in the population. Mutations occur as a Poisson process of rate ρ along the branches of the coalescent tree. The allelic partition groups together individuals of the same allelic type, and is obtained by tracing each individual’s lineage back in time to the most recent mutation. An example of this construction is given below.