Large-dimensional Central Limit Theorem with Fourth-moment Error Bounds on Convex Sets and Balls

Understand

We prove the large-dimensional Gaussian approximation of a sum of $n$ independent random vectors in $\mathbb{R}^d$ together with fourth-moment error bounds on convex sets and Euclidean balls.

• We show that compared with classical third-moment bounds, our bounds have near-optimal dependence on $n$ and 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 $d=o(n)$.
• We discuss an application to the bootstrap.