The Berry-Esseen Constant Conjecture is Eventually True

Abstract

We prove that the Esseen's constant $0.4097...$ is valid 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\left\lceil\exp(10^{17})\right\rceil. $