Quantitative diffusion approximation for the Neutral $r$-Alleles Wright-Fisher Model with Mutations

TL;DR

This paper develops a diffusion approximation for the Wright-Fisher model with multiple neutral alleles, providing improved error bounds that depend on mutation rates and population size, extending previous work from the two-allele case.

Contribution

It introduces a new approximation method using a Lindeberg principle for multiple alleles, improving error bounds over prior two-allele results.

Findings

Error rate depends linearly on maximum mutation rate and inversely on population size.

The approximation extends previous results from two alleles to r alleles.

Provides a quantitative measure of approximation accuracy in the Wright-Fisher model.

Abstract

We apply a Lindeberg principle under the Markov process setting to approximate the Wright-Fisher model with neutral $r$-alleles using a diffusion process, deriving an error rate based on a function class distance involving fourth-order bounded differentiable functions. This error rate consists of a linear combination of the maximum mutation rate and the reciprocal of the population size. Our result improves the error bound in the seminal work [PNAS,1977], where only the special case $r=2$ was studied.