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- W2950356257 abstract "Consider a random permutation of ${1, ldots, lfloor n^{t_2}rfloor}$ drawn according to the Ewens measure with parameter $t_1$ and let $K(n, t)$ denote the number of its cycles, where $tequiv (t_1, t_2)inmathbb [0, 1]^2$. Next, consider a sample drawn from a large, neutral population of haploid individuals subject to mutation under the infinitely many sites model of Kimura whose genealogy is governed by Kingman's coalescent. Let $S(n, t)$ count the number of segregating sites in a sample of size $lfloor n^{t_2}rfloor$ when mutations arrive at rate $t_1/2$. We show that $K(n, (t_1/log n, t_2))-1$ and $S(n, (t_1/log n, t_2))$ induce unique random measures $Pi_n^K$ and $Pi_n^S,$ respectively, on the positive quadrant $[0, infty)^2.$ Our main result is to show that in the coupling of $S(n, t)$ and $K(n, t)$ introduced in~cite{Pitters2019} we have weak convergence as $ntoinfty$ begin{align*} (Pi_n^K, Pi_n^S)to_d (Pi, Pi), end{align*} where $Pi$ is a Poisson point process on $[0, infty)^2$ of unit intensity. This complements the work in~cite{Pitters2019} where it was shown that the process ${(K(n, t), S(n, t)), tin [0, 1]^2},$ appropriately rescaled, converges weakly to the product of the same one-dimensional Brownian sheet." @default.
- W2950356257 created "2019-06-27" @default.
- W2950356257 creator A5018139184 @default.
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- W2950356257 date "2019-06-14" @default.
- W2950356257 modified "2023-09-27" @default.
- W2950356257 title "Poisson limit for the number of cycles in a random permutation and the number of segregating sites" @default.
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