Asymptotics of the allele frequency spectrum and the number of alleles

Abstract

We derive large-sample and other limiting distributions of components of the allele frequency spectrum vector, Kn, joint with the number of alleles, Kn, from a sample of n genes. Models analysed include those constructed from gamma and α-stable subordinators by Kingman (thus including the Ewens model), the two-parameter extension by Pitman and Yor, and a two-parameter version constructed by omitting large jumps from an α-stable subordinator. In each case the limiting distribution of a finite number of components of Kn is derived, joint with Kn. New results include that in the Poisson- Dirichlet case, Kn and Kn are asymptotically independent after centering and norming for Kn, and it is notable, especially for statistical applications, that in other cases the limiting distribution of a finite number of components ofKn, after centering and an unusual nα/2 norming, conditional on that of Kn, is normal. Show more