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

Abstract

For each $N\geqq 1$, let $\{ {X_n^N ,n\geq 0} \}$ be a discrete-time stochastic process, and let $\Delta X_n^N = X_{n + 1}^N - X_n^N $. Suppose that $E( {\Delta X_n^N | {X_n^N } } ) = O( {\varepsilon ^N } )$ and $\operatorname{var} ( {\Delta X_n^N | {X_n^N } } ) = O( {\tau ^N } )$, where $\varepsilon ^N \to 0$ and ${\tau ^N / \varepsilon ^N \to 0}$ as $N \to \infty $. Conditions are given under which there are constants $\gamma _n^N $ such that $Z_n^N ( {X_n^N - \gamma _n^N } )( \varepsilon ^N / \tau ^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 $\gamma _n^N $ are the gene frequencies specified by Haldane’s deterministic theory of evolution.