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

DOI10.1214/22-AOS2184arXiv2009.06004WikidataQ115021750 ScholiaQ115021750MaRDI QIDQ6348990

Miles E. Lopes

Publication date: 13 September 2020

Abstract: 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.

Mathematics Subject Classification ID

Central limit and other weak theorems (60F05) Approximations to statistical distributions (nonasymptotic) (62E17)

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