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Diffusion approximations of Markov chains with two time scales and applications to population genetics, II

Published online by Cambridge University Press:  01 July 2016

S. N. Ethier  and
Thomas Nagylaki
S. N. Ethier*
Affiliation:
University of Utah
Thomas Nagylaki*
Affiliation:
University of Chicago
*
∗Postal address: Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA.
∗∗Postal address: Department of Molecular Genetics and Cell Biology, The University of Chicago, 920 East 58th Street, Chicago, IL 60637, USA.
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Abstract

For N = 1, 2, …, let {(XN (k), YN (k)), k = 0, 1, …} be a time-homogeneous Markov chain in . Suppose that, asymptotically as N → ∞, the ‘infinitesimal' covariances and means of XN([·/ε N]) are aij(x, y) and bi(x, y), and those of YN ([·/δ N]) are 0 and cl(x, y). Assume and limN→∞ε N/δ N = 0. Then, under a global asymptotic stability condition on dy/dt = c(x, y) or a related difference equation (and under some technical conditions), it is shown that (i) XN([·/ε N]) converges weakly to a diffusion process with coefficients aij(x, 0) and bi(x, 0) and (ii) YN([t/ε N]) → 0 in probability for every t > 0. The assumption in Ethier and Nagylaki (1980) that the processes are uniformly bounded is removed here.

The results are used to establish diffusion approximations of multiallelic one-locus stochastic models for mutation, selection, and random genetic drift in a finite, panmictic, diploid population. The emphasis is on rare, severely deleterious alleles. Models with multinomial sampling of genotypes in the monoecious, dioecious autosomal, and X-linked cases are analyzed, and an explicit formula for the stationary distribution of allelic frequencies is obtained.

Keywords

DIFFUSION APPROXIMATIONSMARTINGALE PROBLEMRANDOM GENETIC DRIFTHARDY–WEINBERG PROPORTIONSRARE ALLELES
Type
Research Article
Information
Advances in Applied Probability , Volume 20 , Issue 3 , September 1988 , pp. 525 - 545
Copyright
Copyright © Applied Probability Trust 1988 

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Footnotes

Supported in part by NSF grant DMS-8403648.

Supported in part by NSF grant BSR-8512844.

†

In general, Ez[f(ZN(1))] denotes the integral of f with respect to the one-step transition function of ZN(·) starting at z; a similar convention applies to probabilities, variances, and covariances.

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