Annals of Applied Probability

Padé approximants and exact two-locus sampling distributions

Paul A. Jenkins and Yun S. Song

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For population genetics models with recombination, obtaining an exact, analytic sampling distribution has remained a challenging open problem for several decades. Recently, a new perspective based on asymptotic series has been introduced to make progress on this problem. Specifically, closed-form expressions have been derived for the first few terms in an asymptotic expansion of the two-locus sampling distribution when the recombination rate ρ is moderate to large. In this paper, a new computational technique is developed for finding the asymptotic expansion to an arbitrary order. Computation in this new approach can be automated easily. Furthermore, it is proved here that only a finite number of terms in the asymptotic expansion is needed to recover (via the method of Padé approximants) the exact two-locus sampling distribution as an analytic function of ρ; this function is exact for all values of ρ ∈ [0, ∞). It is also shown that the new computational framework presented here is flexible enough to incorporate natural selection.

Article information

Ann. Appl. Probab., Volume 22, Number 2 (2012), 576-607.

First available in Project Euclid: 2 April 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 92D15: Problems related to evolution
Secondary: 65C50: Other computational problems in probability 92D10: Genetics {For genetic algebras, see 17D92}

Population genetics recombination sampling distribution asymptotic expansion Padé approximants


Jenkins, Paul A.; Song, Yun S. Padé approximants and exact two-locus sampling distributions. Ann. Appl. Probab. 22 (2012), no. 2, 576--607. doi:10.1214/11-AAP780.

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