Poisson limit for the number of cycles in a random permutation and the number of segregating sites

• Main Authors: Pitters, Helmut H, Weissmann, Philip
• Format: Journal Article
• Language: English
• Published: 14.06.2019
• Subjects: Mathematics - Probability
• DOI: 10.48550/arxiv.1906.06336

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$t\equiv (t_1, t_2)\in\mathbb [0, 1]^2$ . 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~Pitters2019 we have weak convergence as$n\to\infty$align* (_n^K, _n^S)_d ( ), align* where$\Pi$is a Poisson point process on$[0, \infty)^2$of unit intensity. This complements the work in~Pitters2019 where it was shown that the process$\{(K(n, t), S(n, t)), t\in [0, 1]^2\},$appropriately rescaled, converges weakly to the product of the same one-dimensional Brownian sheet.