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

09/01/2020

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by   Xiao Fang, et al.

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We prove the large-dimensional Gaussian approximation of a sum of n independent random vectors in 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 can achieve improved and, in the case of balls, optimal dependence d=o(n) on dimension. We discuss an application to the bootstrap. The proof is by recent advances in Stein's method.