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The multivariate Berry-Esseen theorem with an absolute constant
Abstract
We prove that the multivariate Berry--Esseen inequality for sums of independent random vectors holds with the same constant in every dimension. For independent centered random vectors X1,…,Xn∈ℝd whose covariance matrices sum to Id, we show that the distance between the distribution of their sum and N(0,Id), measured uniformly over convex sets, is at most C0∑i=1n𝔼|Xi|23 for a universal constant C0. The bound requires only finite third moments and holds uniformly in the dimension, the number of summands, and their distributions. It also yields the corresponding bound for independent and identically distributed vectors. In particular, C0 can be chosen to be $980$.
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BibTeX
@misc{ling2026multivariate,
title = {{The multivariate Berry-Esseen theorem with an absolute constant}},
author = {Zhi Ling},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02195},
eprint = {2611.02195},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02195},
}
RIS
TY - UNPB
TI - The multivariate Berry-Esseen theorem with an absolute constant
AU - Zhi Ling
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.02195
UR - https://arxiv.org/abs/2611.02195
AB - We prove that the multivariate Berry--Esseen inequality for sums of independent random vectors holds with the same constant in every dimension. For independent centered random vectors X_1,…,X_n∈ℝ^d whose covariance matrices sum to I_d, we show that the distance between the distribution of their sum and N(0,I_d), measured uniformly over convex sets, is at most C_0∑_i=1^n𝔼|X_i|_2^3 for a universal constant C_0. The bound requires only finite third moments and holds uniformly in the dimension, the number of summands, and their distributions. It also yields the corresponding bound for independent and identically distributed vectors. In particular, C_0 can be chosen to be $980$.
ER -