Approximation of Stochastic Processes by Gaussian Diffusions, and Applications to Wright-Fisher Genetic Models

For each N 1, let \ XₙN ,n≥ 0 \ be a discrete-time stochastic process, and let Δ XₙN = Xn + 1N - XₙN. Suppose that E( Δ XₙN | XₙN ) = O( ε N ) and var ( Δ XₙN | XₙN ) = O( τ N ), where ε N → 0 and τ N / ε N → 0 as N → ∞. Conditions are given under which there are constants γ ₙN such that ZₙN ( XₙN - γ ₙN )( ε N / τ N )1 / 2 can be approximated by a Gaussian diffusion when N is large. It is shown that these conditions are satisfied by the Wright–Fisher models for fluctuations in gene frequency under theinfluence of mutation, selection and random drift. For these models, N is the population size and the constants γ ₙN are the gene frequencies specified by Haldane’s deterministic theory of evolution.