On combinatorial central limit theorems with different underlying permutations via approximate zero biasing

We provide new bounds between a combinatorial statistic of the form  and the standard normal distribution, where  are independent real valued random variables, and , which is independent of  and follows either the uniform or Ewens distribution. The family of the Ewens distributions appears in the context of population genetics in biology. The bounds are based on both  and  distances under different assumptions. As our method, we apply the approximate zero bias approach via Stein’s method to obtain the bounds.