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arXiv · 2609.06358

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

Hengzhi He·Guang Cheng
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Abstract

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

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Hengzhi He, Guang Cheng. 2026-09-06. The Berry--Esseen Bound is Sharp for All Sufficiently Large Sample Sizes. https://arxiv.org/abs/2609.06358

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