The Berry--Esseen Bound is Sharp for All Sufficiently Large Sample Sizes

Abstract

We prove that the Esseen's constant $\mathrm{0.4097...}$ is sharp in the iid Berry--Esseen inequality as long as the sample sizes $n\ge N$. The threshold $N$ is universal over all real summand distributions with positive variance and a finite third absolute moment. In particular, one may take $N=2\lceil \exp (10^{17})\rceil .$