Large-dimensional central limit theorem with fourth-moment error bounds on convex sets and balls

Abstract

We prove the large-dimensional Gaussian approximation of a sum of n independent random vectors in ${\mathbb{R}}^{\mathit{d}}$ together with fourth-moment error bounds on convex sets and Euclidean balls. Our bounds have near-optimal dependence on n and, compared with classical third-moment bounds, can achieve improved dependence on the dimension d. For centered balls, we obtain an additional error bound that has a sub-optimal dependence on n, but recovers the known result of the validity of the Gaussian approximation if and only if $\mathit{d}=\mathit{o}(\mathit{n})$. We discuss an application to the bootstrap. We prove our main results using Stein’s method.