An optimal Berry-Esseen type inequality for expectations of smooth functions

Abstract

We provide an optimal Berry-Esseen type inequality for Zolotarev’s ideal ζ₃-metric measuring the difference between expectations of sufficiently smooth functions, like |·|³, of a sum of independent random variables X₁,..., X_n with finite third-order moments and a sum of independent symmetric two-point random variables, isoscedastic to the X_i. In the homoscedastic case of equal variances, and in particular, in case of identically distributed X₁,..., X_n the approximating law is a standardized symmetric binomial one. As a corollary, we improve an already optimal estimate of the accuracy of the normal approximation due to Tyurin (2009).