Optimal central limit theorem for bounded random variables in high dimensions

Abstract

Let $W=n^{-1/2}{\sum }_{i=1}^n{X}_i$, where the $X_i$ are independent centered random vectors in ${\mathbb{R}}^p$ with $|X_{ij}|\le B$ almost surely. Suppose that $\text{Cov}(W)$ has unit diagonal and smallest eigenvalue at least $b^2>0$. We prove that the distance between $W$ and a Gaussian vector with the same covariance, uniformly over axis-aligned rectangles, is at most $C\min \{1,b^{-2}B{n}^{-1/2}{\log }^{3/2}(ep)\}$. For fixed $b$, the dependence on summand size and dimension matches known lower bounds in growing-dimensional regimes. The proof combines a concentration estimate near rectangle boundaries with a carefully chosen Gaussian comparison.