Notes on the dimension dependence in high-dimensional central limit theorems for hyperrectangles

摘要

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) Σ_(i=1)~n X_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 (logd)~5/n→0. This improves a result of Chernozhukov et al. (Ann Probab 45:2309-2353, 2017) in terms of the dimension growth condition. When n~(-1/2) Σ_(i=1)~n X_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.