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 Electronic Journal of Probability > Vol. 13 (2008) > Paper 17 open journal systems 


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.


Full text: PDF

Pages: 486-512

Published on: March 31, 2008
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Electronic Journal of Probability. ISSN: 1083-6489