Poisson Process Approximations for the Ewens Sampling Formula

Abstract

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 θ = 1. Under the Ewens sampling formula with parameter θ, the process of cycle counts (C 1 (n), C 2 (n), ..., C n (n), 0, 0, ...) converges to a Poisson process (Z 1, Z 2, ...) with independent coordinates and EZ j = θ/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), ..., C b (n)) and (Z 1, ..., 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.