The Independent Berry–Esseen Constant

Current state

For independent centered random variables with total variance one and finite third absolute moments, write $L={\sum }_i\mathbb{E}|X_i{|}^3$. Let $\Delta$ be the largest absolute difference between the distribution function of their sum and the standard normal distribution function. The independent Berry–Esseen constant is the least $C_{\mathrm{ind}}$ for which $\Delta \le C_{\mathrm{ind}}L$ holds for every such finite family.

The upper bound proved in this collection gives

Esseen (1956) established the lower bound. The upper bound combines analytic estimates with a certificate checked using interval arithmetic.

We prove the sharp bound $\Delta \le C_{\mathrm{E}}L$ for arbitrary independent summands when $L\le \exp (-\exp (20004))$. This argument extends a method of He and Cheng (2026) from identically distributed samples to independent arrays. The sharp bound also holds for Bernoulli sums whose supports have a common diameter, and for three Bernoulli summands with arbitrary support diameters. For two arbitrary summands, we obtain the stronger bound $\Delta <(49/120)L$.

A fourth paper studies extremal distributions and proves necessary conditions under a bound on the sum of third absolute moments. The collection includes all four manuscripts, their source files, and verification code.

Future plans

The conjecture $C_{\mathrm{ind}}=C_{\mathrm{E}}$ remains open. The goal is to prove this equality and, along the way, tighten the unconditional upper bound.