Refined Approximations for the Ewens Sampling Formula
A. D. Barbour · Random Structures and Algorithms · 1992
Abstract The Ewens sampling formula is a family of probability distributions over the space of cycle types of permutations ofnobjects, indexed by a real parameter θ. In the case θ = 1, where the distribution reduces to that induced by the uniform distribution on all permutations, the joint distributions of the numbers of cycles of lengths less thanb = o(n)is extremely well approximated by a product of Poisson distributions, having mean 1/jfor cycle lengthj: the error is super‐exponentially small withnb−1. For θ ≠ 1. the analogous approximation, with means adjusted to θ/j, is good, but with error only linear inn−1b. In this article, it is shown that, by choosing the means of the Poisson distributions more carefully, an error quadratic inn−1bcan be achieved, and that essentially nothing better is possible.