Notes on the dimension dependence in high-dimensional central limit theorems for hyperrectangles
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Let X_1,...,X_n be independent centered random vectors in R^d. This paper shows that, even when d may grow with n, the probability P(n^-1/2∑_i=1^nX_i∈ A) can be approximated by its Gaussian analog uniformly in hyperrectangles A in R^d as n→∞ under appropriate moment assumptions, as long as (log d)^5/n→0. This improves a result of Chernozhukov, Chetverikov & Kato [Ann. Probab.45 (2017) 2309–2353] in terms of the dimension growth condition. When n^-1/2∑_i=1^nX_i has a common factor across the components, this condition can be further improved to (log d)^3/n→0. The corresponding bootstrap approximation results are also developed. These results serve as a theoretical foundation of simultaneous inference for high-dimensional models.
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