Steady-state Dirichlet approximation of the Wright-Fisher model using the prelimit generator comparison approach of Stein's method

Abstract

The Wright-Fisher model, originating in Wright (1931) is one of the canonical probabilistic models used in mathematical population genetics to study how genetic type frequencies evolve in time. In this paper we bound the rate of convergence of the stationary distribution for a finite population Wright-Fisher Markov chain with parent independent mutation to the Dirichlet distribution. Our result improves the rate of convergence established in Gan et al. (2017) from $O(1/\sqrt{N})$ to $O(1/N)$. The results are derived using Stein's method, in particular, the prelimit generator comparison method.