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

For N = 1, 2, ···, let {( X N ( k ), Y N ( k )), k = 0, 1, ···} be a homogeneous Markov chain in ℝ m x ℝ n . Suppose that, asymptotically as N → ∞, the ‘infinitesimal’ covariances and means of X N ([·/∊ N ]) are a ij ( x, y ) and b i ( x, y ), and those of Y N ([·/δ N ]) are 0 and c l ( x, y ). Assume lim N →∞ δ N = lim N →∞ ∊ N /δ N = 0 and the zero solution of ý = c ( x, y ) is globally asymptotically stable. Then, under some technical conditions, it is shown that (i) X N ([·/∊ N ]) converges weakly to a diffusion process with coefficients a ij ( x , 0) and b i ( x , 0), and (ii) Y N ([ t /∊ N ]) → 0 in probability for every t > 0. (The case lim N →∞ δ N = δ ∞ > 0 = lim N →∞ ∊ N is also treated.) The proof is based on the discrete-parameter analogue of a generalization of Kurtz's limit theorems for perturbed operator semigroups. The results are applied to three classes of stochastic models for random genetic drift at a multiallelic locus in a finite diploid population. The three classes involve multinomial sampling, overlapping generations, and general progeny distributions. Within each class, the monoecious, dioecious autosomal, and X-linked cases are analyzed. It is found that results for a monoecious population obtained from a diffusion approximation can be applied at once to the dioecious cases by using the appropriate effective population size and averaging allelic frequencies, selection intensities, and mutation rates, weighting each sex by the number of genes carried by an individual at the locus under consideration.