Poisson Process Approximations for the Ewens Sampling Formula

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 .>0; the usual uniform measure is included as the special case .=1. Under the Ewens sampling formula with parameter ., the process of cycle counts (C1(n),C2(n),.,Cn(n),0,0,.) converges to a Poisson process (Z1,Z2,.) with independent coordinates and EZj=./j. Exploiting a particular coupling, we give simple explicit upper bounds for the Wasserstein and total variation distances between the laws of (C1(n),.,Cb(n)) and (Z1,.,Zb). 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.