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

  • S. N. Ethier (a1) and Thomas Nagylaki (a2)
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 lim N→∞ ε NN = 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.

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Corresponding author
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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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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Advances in Applied Probability
  • ISSN: 0001-8678
  • EISSN: 1475-6064
  • URL: /core/journals/advances-in-applied-probability
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