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

Abstract

ForN =1, 2, …, let {(XN(k),YN(k)),k =0, 1, …} be a time-homogeneous Markov chain in. Suppose that, asymptotically asN → ∞, the ‘infinitesimal' covariances and means ofXN([·/εN]) areaij(x, y) andbi(x, y), and those ofYN([·/δN]) are 0 andcl(x, y). Assumeand limN→∞εN/δN= 0. Then, under a global asymptotic stability condition ondy/dt = c(x, y) or a related difference equation (and under some technical conditions), it is shown that (i)XN([·/εN]) converges weakly to a diffusion process with coefficientsaij(x, 0) andbi(x, 0) and (ii)YN([t/εN]) → 0 in probability for everyt> 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, andX-linked cases are analyzed, and an explicit formula for the stationary distribution of allelic frequencies is obtained.