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A stability property of the Ewens sampling formula

Published online by Cambridge University Press:  14 July 2016

Stanley Sawyer
Stanley Sawyer*
Affiliation:
Purdue University
*
∗ Permanent address: Department of Mathematics, Purdue University, W. Lafayette, IN 47907, U.S.A.
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Abstract

An error bound for convergence to the Ewens sampling formula is given where the population size or mutation rate may vary from generation to generation, or the population is not yet at equilibrium. An application is given to a model of Hartl and Campbell about selectively-equivalent subtypes within a class of deleterious alleles, and a theorem is proven showing that the size of the deleterious class stays within bounds sufficient to apply the first result. Generalizations are discussed.

Keywords

SAMPLINGNEUTRAL SAMPLINGGAUSSIAN APPROXIMATIONALLELISMDELETERIOUS ALLELES
Type
Research Papers
Information
Journal of Applied Probability , Volume 20 , Issue 3 , September 1983 , pp. 449 - 459
Copyright
Copyright © Applied Probability Trust 1983 

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Footnotes

Partially supported by NSF Grant MCS82–02858.

References

Crow, J. and Kimura, M. (1970) An Introduction to Population Genetics Theory. Harper and Row, New York.Google Scholar
Ethier, S. and Kurtz, T. (1981) The infinitely-many-neutral-alleles diffusion model. Adv. Appl. Prob. 13, 429–452.Google Scholar
Ethier, S. and Nagylaki, T. (1980) Diffusion approximations of Markov chains with two time scales and applications to population genetics. Adv. Appl. Prob. 12, 14–49.Google Scholar
Ewens, W. (1972) The sampling theory of selectively neutral alleles. Theoret. Popn Biol. 3, 87–112.CrossRefGoogle ScholarPubMed
Ewens, W. (1979) Mathematical Population Genetics. Biomathematics Vol. 9, Springer-Verlag, New York.Google Scholar
Ewens, W. and Kirby, K. (1975) The eigenvalues of the neutral alleles process. Theoret. Popn Biol. 7, 212–220.Google Scholar
Griffiths, R. C. (1979) Exact sampling distributions from the infinite neutral alleles model. Adv. Appl. Prob. 11, 326–354.Google Scholar
Hartl, D. L. and Campbell, R. (1982) Allele multiplicity in simple Mendelian disorders. Amer. J. Hum. Genet. 34, 866–873.Google ScholarPubMed
Karlin, S. and Avni, H. (1975) Derivation of the eigenvalues of the configuration process induced by a labeled direct product branching process. Theoret. Popn Biol. 7, 221–228.CrossRefGoogle ScholarPubMed
Karlin, S. and Mcgregor, J. (1972) Addendum to a paper of W. Ewens. Theoret. Popn Biol. 3, 112–114.CrossRefGoogle ScholarPubMed
Kurtz, T. (1971) Limit theorems for sequences of jump Markov processes approximating ordinary differentiable processes. J. Appl. Prob. 8, 344–356.Google Scholar
Kurtz, T. (1981) Approximation of population processes. CBMS-NSF Regional Conference Series in Applied Mathematics 36.Google Scholar
Nagylaki, T. (1977) Selection in One- and Two-locus Systems. Lecture Notes in Biomathematics 15, Springer-Verlag, Berlin.CrossRefGoogle ScholarPubMed
Norman, F. (1975) Approximation of stochastic processes by Gaussian diffusions, and applications to Wright-Fisher genetic models. SIAM J. Appl. Math. 29, 225–242.CrossRefGoogle Scholar
Sawyer, S. and Hartl, D. L. (1981) On the evolution of behavioral reproductive isolation: the Wallace effect. Theoret. Popn Biol. 19, 261–273.Google Scholar