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

Part of: Special processes Limit theorems

Published online by Cambridge University Press:  27 May 2025

Anton Braverman Open the ORCID record for Anton Braverman [Opens in a new window]  and
Han Liang Gan Open the ORCID record for Han Liang Gan [Opens in a new window]
Anton Braverman*
Affiliation:
Northwestern University
Han Liang Gan*
Affiliation:
University of Waikato
*
*Postal address: Kellogg School of Management, 2211 Campus Drive, Evanston, IL 60208, USA. Email: anton.braverman@kellogg.northwestern.edu
**Postal address: Department of Mathematics, Private Bag 3105, Hamilton 3240, New Zealand. Email: han.gan@waikato.ac.nz
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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 $\mathrm{O}(1/\sqrt{N})$ to $\mathrm{O}(1/N)$. The results are derived using Stein’s method, in particular, the prelimit generator comparison method.

Keywords

Dirichlet approximationWright–Fisher modelStein’s method

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 60K99: None of the above, but in this section
Type
Original Article
Information
Advances in Applied Probability , First View , pp. 1 - 39
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Applied Probability Trust

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