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, 𝐌𝑛, 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 𝐌𝑛 is derived, joint with 𝐾𝑛. New results include that in the Poisson–Dirichlet case, 𝐌𝑛 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 of 𝐌𝑛, after centering and an unusual 𝑛𝛼/2 norming, conditional on that of 𝐾𝑛, is normal.