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Central Limit Theorem and Bootstrap Approximation in High Dimensions: Near $1/\sqrt{n}$ Rates via Implicit Smoothing

Non-asymptotic bounds for Gaussian and bootstrap approximation have recently attracted significant interest in high-dimensional statistics. This paper studies Berry-Esseen bounds for such approximations with respect to the multivariate Kolmogorov distance, in the context of a sum of $n$ random vectors that are $p$-dimensional and i.i.d. Up to now, a growing line of work has established bounds with mild logarithmic dependence on $p$. However, the problem of developing corresponding bounds with near $n^{-1/2}$ dependence on $n$ has remained largely unresolved. Within the setting of random vectors that have sub-Gaussian or sub-exponential entries, this paper establishes bounds with near $n^{-1/2}$ dependence, for both Gaussian and bootstrap approximation. In addition, the proofs are considerably distinct from other recent approaches and make use of an "implicit smoothing" operation in the Lindeberg interpolation.