The multivariate Berry-Esseen theorem with an absolute constant
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Zhi Ling · arXiv: 2611.02195v1 · Summary page
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Normal approximation without a dimension-dependent constant
The central limit theorem explains why sums of many independent contributions often look Gaussian. Berry–Esseen bounds make that approximation quantitative. A new preprint claims a multivariate bound with a universal constant: the same constant works regardless of the dimension, the number of contributions or their individual distributions, provided the stated covariance normalization and finite third-moment assumptions hold.
The discrepancy is measured over all convex sets, a broad family of events relevant to multivariate inference. The error bound depends on the sum of the contributions' expected cubed Euclidean lengths. That distinction matters: eliminating dimension from the constant does not eliminate it from the quantities entering the bound. For researchers using Gaussian approximations, the proposed theorem offers a cleaner general guarantee rather than an assurance that high-dimensional approximation is always accurate.
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