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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: ma.ethier@science.utah.edu.
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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

Donnelly, P. (1986) Partition structures, Polya urns, the Ewens sampling formula and the ages of alleles. Theoret. Popn. Biol. 30, 271–288.Google Scholar
Donnelly, P. and Joyce, P. (1989) Continuity and weak convergence of ranked and size-biased permutations on the infinite simplex. Stoch. Proc. Appl. 31, 89–103.Google Scholar
Donnelly, P. and Joyce, P. (1991) Consistent ordered sampling distributions: characterization and convergence, Adv. Appl. Prob. 23 (2)Google Scholar
Donnelly, P. and Tavaré, S. (1986) The ages of alleles and a coalescent. Adv. Appl. Prob. 18, 1–19. Correction 18, 1023.Google Scholar
Donnelly, P. and Tavaré, S. (1987) The population genealogy of the infinitely-many neutral alleles model. J. Math. Biol. 25, 381–391.Google Scholar
Ethier, S. N. (1990) The infinitely-many-neutral-alleles diffusion model with ages. Adv. Appl. Prob. 22, 1–24.Google Scholar
Ethier, S. N. and Kurtz, T. G. (1981) The infinitely-many-neutral-alleles diffusion model. Adv. Appl. Prob. 13, 429–452.CrossRefGoogle Scholar
Ewens, W. J. (1990) Population genetics theory—the past and the future. In Mathematical and Statistical Developments of Evolutionary Theory, ed. Lessard, S., Reidel, Dordrecht, pp. 177–227.Google Scholar
Fleming, W. H. and Viot, M. (1979) Some measure-valued Markov processes in population genetics theory. Indiana Univ. Math. J. 28, 817–843.CrossRefGoogle Scholar
Hoppe, F. M. (1984) Pólya-like urns and the Ewens sampling formula. J. Math. Biol. 20, 91–94.Google Scholar
Hoppe, F. M. (1987) The sampling theory of neutral alleles and an urn model in population genetics. J. Math. Biol. 25, 123–159.Google Scholar
Kelly, F. P. (1979) Reversibility and Stochastic Networks. Wiley, Chichester.Google Scholar
Kingman, J. F. C. (1975) Random discrete distributions. J.R. Statist. Soc. B 37, 1–22.Google Scholar
Littler, R. A. and Good, A. J. (1978) Ages, extinction times, and first passage probabilities for a multiallele diffusion model with irreversible mutation. Theoret. Popn. Biol. 13, 214–225.Google Scholar
Patil, G. P. and Taillie, C. (1977) Diversity as a concept and its implications for random communities. Bull, Internat. Statist. Inst. 47, 497–515.Google Scholar
Watterson, G. A. and Guess, H. A. (1977) Is the most frequent allele the oldest? Theoret. Popn. Biol. 11, 141–160.Google Scholar