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On the past history of an allele now known to have frequency p

Published online by Cambridge University Press:  14 July 2016

Stanley Sawyer
Stanley Sawyer*
Affiliation:
Yeshiva University, New York
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Abstract

Consider a diploid population of N individuals undergoing random mating and mutation as in the infinite-allele Wright model. Choose a particular gene in generation Q for large Q, and let ft (0 ≦ t ≦ Q) be the frequency of the allelic type of the predecessor of that particular gene (or the gene itself) in generation t. By considering a ‘diffusion approximation' {xt} of {ft}, we find the distribution of the age of an allelic type now known to have frequency p, and of its distribution of frequencies since the allele came into existence. A novelty here is that the process {xt} is not a diffusion, but a process with jumps; it has x = 0 as an inaccessible entrance boundary but periodically jumps to it from the interior of [0, 1]. The formulas obtained are the same as those derived by Maruyama and Kimura, who used a totally different approach.

Keywords

GENETICSPOPULATION GENETICSAGE OF AN ALLELESELECTIVE NEUTRALITYWRIGHT MODELKIMURA–CROW MODELRETURN PROCESS
Type
Research Papers
Information
Journal of Applied Probability , Volume 14 , Issue 3 , September 1977 , pp. 439 - 450
Copyright
Copyright © Applied Probability Trust 1977 

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