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, $\mathbf{M}_n$ , joint with the number of alleles, $K_n$ , from a sample of n genes. Models analysed include those constructed from gamma and $\alpha$ -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 $\alpha$ -stable subordinator. In each case the limiting distribution of a finite number of components of $\mathbf{M}_n$ is derived, joint with $K_n$ . New results include that in the Poisson–Dirichlet case, $\mathbf{M}_n$ and $K_n$ are asymptotically independent after centering and norming for $K_n$ , and it is notable, especially for statistical applications, that in other cases the limiting distribution of a finite number of components of $\mathbf{M}_n$ , after centering and an unusual $n^{\alpha/2}$ norming, conditional on that of $K_n$ , is normal.