Weak convergence of discrete time non-markovian processes related to selection models in population genetics

We consider discrete time stochastic processes defined by solutions to some non-linear difference equations whose coefficients are autocorrelated random sequences. It is proved that these processes converge weakly in D[0, T] to diffusion processes, under the assumption that the random sequences satisfy some mixing condition. Diffusion approximation for stochastic selection models in population genetics is discussed, as the application of this limit theorem.