Asymptotics of the allele frequency spectrum associated with the Bolthausen-Sznitman coalescent
Anne-Laure Basdevant, Université Paul Sabatier (Toulouse III)
Christina Goldschmidt, Department of Statistics, University of Oxford
Abstract
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
(N1(n), N2(n),..., Nn(n)), where Nk(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) N1(n) converges in probability to rho and, for k at
least 2, n-1(log n)2 Nk(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.
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