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Asymptotics of the allele frequency spectrum and the number of alleles
- Part of
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- Journal:
- Journal of Applied Probability , First View
- Published online by Cambridge University Press:
- 22 November 2024, pp. 1-25
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- Article
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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.
