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

For N = 1, 2, ., let {(XN (k), YN (k)), k = 0, 1, .} be a time-homogeneous Markov chain in https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:bina ry:20180209072205981-0599:S0001867800018139:S0001867800018139_inline1.gi f?pub-status=live . Suppose that, asymptotically as N . ., the .infinitesimal' covariances and means of XN([·/. N]) are aij(x, y) and bi(x, y), and those of YN ([·/. N]) are 0 and cl(x, y). Assume https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:bina ry:20180209072205981-0599:S0001867800018139:S0001867800018139_inline2.gi f?pub-status=live and limN... N/. 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([·/. N]) converges weakly to a diffusion process with coefficients aij(x, 0) and bi(x, 0) and (ii) YN([t/. 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.