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Asymptotics of the allele frequency spectrum and the number of alleles

Part of: Applications Stochastic processes Distribution theory

Published online by Cambridge University Press:  22 November 2024

Ross A. Maller Open the ORCID record for Ross A. Maller [Opens in a new window]  and
Soudabeh Shemehsavar
Ross A. Maller*
Affiliation:
The Australian National University
Soudabeh Shemehsavar*
Affiliation:
Murdoch University and University of Tehran
*
*Postal address: Research School of Finance, Actuarial Studies and Statistics, Australian National University, Canberra, ACT, 0200, Australia. Email address: Ross.Maller@anu.edu.au
**Postal address: College of Science, Technology, Engineering and Mathematics, and Centre for Healthy Ageing, Health Future Institute, Murdoch University, and School of Mathematics, Statistics & Computer Sciences, University of Tehran. Email address: Soudabeh.Shemehsavar@murdoch.edu.au
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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.

Keywords

Allele frequency spectrumgeneralised Poisson–Dirichlet lawsEwens sampling formulaKingman’s Poisson–Dirichlet distributionsPitman sampling formulagene and species distributions

MSC classification

Primary: 60G51: Processes with independent increments; L'evy processes 60G52: Stable processes 60G55: Point processes
Secondary: 60G57: Random measures 62E20: Asymptotic distribution theory 62P10: Applications to biology and medical sciences
Type
Original Article
Information
Journal of Applied Probability , Volume 62 , Issue 2 , June 2025 , pp. 516 - 540
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Applied Probability Trust

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