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We prove the large-dimensional Gaussian approximation of a sum of $n$independent random vectors in $\mathbb{R}^d$ together with fourth-moment errorbounds on convex sets and Euclidean balls. We show that compared with classicalthird-moment bounds, our bounds have near-optimal dependence on $n$ and canachieve improved dependence on the dimension $d$. For centered balls, we obtainan additional error bound that has a sub-optimal dependence on $n$, butrecovers the known result of the validity of the Gaussian approximation if andonly if $d=o(n)$. We discuss an application to the bootstrap. We prove our mainresults using Stein's method.

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