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The distribution of the frequencies of age-ordered alleles in a diffusion model

Published online by Cambridge University Press:  01 July 2016

S. N. Ethier
S. N. Ethier*
Affiliation:
University of Utah
*
∗Postal address: Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA. e-mail address: [email protected].
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Abstract

We prove that the frequencies of the oldest, second-oldest, third-oldest, … alleles in the stationary infinitely-many-neutral-alleles diffusion model are distributed as X1, (1 − X1)X2, (1 − X1)(1 − X2)X3, …, where X1, X2,X3, … are independent beta (1, θ) random variables, θ being twice the mutation intensity; that is, the frequencies of age-ordered alleles have the so-called Griffiths–Engen–McCloskey, or GEM, distribution. In fact, two proofs are given, the first involving reversibility and the size-biased Poisson–Dirichlet distribution, and the second relying on a result of Donnelly and Tavaré relating their age-ordered sampling formula to the GEM distribution.

Keywords

POPULATION GENETICSINFINITELY-MANY-NEUTRAL-ALLELES MODELMEASURE-VALUED DIFFUSIONSPOISSON–DIRICHLET DISTRIBUTIONSIZE BIASINGGEM DISTRIBUTIONORDERED SAMPLING DISTRIBUTIONS
Type
Research Article
Information
Advances in Applied Probability , Volume 22 , Issue 3 , September 1990 , pp. 519 - 532
Copyright
Copyright © Applied Probability Trust 1990 

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Footnotes

Research supported in part by NSF grants DMS-8704369 and DMS-8902991.

References

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