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

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.