Central Limit Theorem and Bootstrap Approximation in High Dimensions with Near 1/√(n) Rates
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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 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.
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