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Asymptotics of the allele frequency spectrum associated with the Bolthausen-Sznitman coalescent. (English) Zbl 1190.60006

Summary: We consider a coalescent process as a model for the genealogy of a sample from a population. The population is subject to neutral mutation at constant rate rho per individual and every mutation gives rise to a completely new type. The allelic partition is obtained by tracing back to the most recent mutation for each individual and grouping together individuals whose most recent mutations are the same. The allele frequency spectrum is the sequence \((N_1(n), N_2(n),\dots, N_n(n))\), where \(N_k(n)\) is number of blocks of size \(k\) in the allelic partition with sample size \(n\). In this paper, we prove law of large numbers-type results for the allele frequency spectrum when the coalescent process is taken to be the Bolthausen-Sznitman coalescent. In particular, we show that \(n^{-1}(\log n) N_{1}(n)\) converges in probability to rho and, for \(k\) at least 2, \(n^{-1}(\log n)^{2} N_{k}(n)\) converges in probability to \(\rho/(k(k-1))\) as \(n\) tends to infinity. Our method of proof involves tracking the formation of the allelic partition using a certain Markov process, for which we prove a fluid limit.

MSC:

60C05 Combinatorial probability
60F05 Central limit and other weak theorems
60J75 Jump processes (MSC2010)
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