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<p>We apply a Lindeberg principle under the Markov process setting to approximate the Wright-Fisher model with neutral <i>r</i>-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 (Ethier and Norman Proc. Natl. Acad. Sci. U. S. A. <b>74</b>(11), 5096–5098 <CitationRef CitationID="CR16">1977</CitationRef>), where only the special case <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(r=2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> was studied.</p>

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Quantitative Diffusion Approximation for the Neutral r-Alleles Wright-Fisher Model with Mutations

  • 期刊论文

  • 发表日期：

    2026年3月19日

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  • Peng Chen,
  • Jie Xiong,
  • Lihu Xu,
  • Jiayu Zheng

摘要

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 (Ethier and Norman Proc. Natl. Acad. Sci. U. S. A. 74(11), 5096–5098 1977), where only the special case \(r=2\) r = 2 was studied.