Open Access
August, 1992 Poisson Process Approximations for the Ewens Sampling Formula
Richard Arratia, A. D. Barbour, Simon Tavare
Ann. Appl. Probab. 2(3): 519-535 (August, 1992). DOI: 10.1214/aoap/1177005647


The Ewens sampling formula is a family of measures on permutations, that arises in population genetics, Bayesian statistics and many other applications. This family is indexed by a parameter $\theta > 0$; the usual uniform measure is included as the special case $\theta = 1$. Under the Ewens sampling formula with parameter $\theta$, the process of cycle counts $(C_1(n), C_2(n), \ldots, C_n(n), 0, 0, \ldots)$ converges to a Poisson process $(Z_1, Z_2, \ldots)$ with independent coordinates and $\mathbb{E}Z_j = \theta/j$. Exploiting a particular coupling, we give simple explicit upper bounds for the Wasserstein and total variation distances between the laws of $(C_1(n), \ldots, C_b(n))$ and $(Z_1, \ldots, Z_b)$. This Poisson approximation can be used to give simple proofs of limit theorems with bounds for a wide variety of functionals of such random permutations.


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Richard Arratia. A. D. Barbour. Simon Tavare. "Poisson Process Approximations for the Ewens Sampling Formula." Ann. Appl. Probab. 2 (3) 519 - 535, August, 1992.


Published: August, 1992
First available in Project Euclid: 19 April 2007

zbMATH: 0756.60006
MathSciNet: MR1177897
Digital Object Identifier: 10.1214/aoap/1177005647

Primary: 60C05
Secondary: 05A05 , 05A16 , 92D10

Keywords: permutations , Population genetics , Total variation

Rights: Copyright © 1992 Institute of Mathematical Statistics

Vol.2 • No. 3 • August, 1992
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