Stein's method using approximate zero bias couplings with applications to combinatorial central limit theorems under the Ewens distribution

TL;DR

This paper extends Stein's method by introducing approximate zero bias couplings and applies it to prove central limit theorems for Ewens distributed permutations, relevant in population genetics.

Contribution

It generalizes zero bias techniques to approximate versions and derives Berry-Esseen bounds for combinatorial CLTs under Ewens distribution.

Findings

Derived Berry-Esseen bounds for normal approximation

Applied bounds to Ewens distribution with $ heta>0$

Extended Stein's method to approximate zero bias couplings

Abstract

We generalize the well-known zero bias distribution and the $\lambda$-Stein pair to an approximate zero bias distribution and an approximate $\lambda,R$-Stein pair, respectively. Berry Esseen type bounds to the normal, based on approximate zero bias couplings and approximate $\lambda,R$-Stein pairs, are obtained using Stein's method. The bounds are then applied to combinatorial central limit theorems where the random permutation has the Ewens $\mathcal{E}_\theta$ distribution with $\theta>0$ which can be specialized to the uniform distribution by letting $\theta=1$. The family of the Ewens distributions appears in the context of population genetics in biology.