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

Abstract

We generalize the well-known zero bias distribution and the $λ$-Stein pair to an approximate zero bias distribution and an approximate $λ,R$-Stein pair, respectively. Berry Esseen type bounds to the normal, based on approximate zero bias couplings and approximate $λ,R$-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 $\mathcal{E}_θ$ distribution with $θ>0$ which can be specialized to the uniform distribution by letting $θ=1$. The family of the Ewens distributions appears in the context of population genetics in biology.