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 $X_1,\ldots,X_n\in\mathbb{R}^d$ 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 $C_0\sum_{i=1}^n\mathbb{E}\|X_i\|_2^3$ 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 $980$.