High-dimensional CLT: Improvements, non-uniform extensions and large deviations

Abstract

Central limit theorems (CLTs) for high-dimensional random vectors with dimension possibly growing with the sample size have received a lot of attention in the recent times. Chernozhukov et al. (Ann. Probab. 45 (2017) 2309–2352) proved a Berry–Esseen type result for high-dimensional averages for the class of sparsely convex sets including hyperrectangles as a special case and they proved that the rate of convergence can be upper bounded by $n^{-1/6}$ up to a polynomial factor of $\log p$ (where $n$ represents the sample size and $p$ denotes the dimension). Convergence to zero of the bound requires $\log^{7}p=o(n)$. We improve upon their result, for hyperrectangles, which only requires $\log^{4}p=o(n)$ (in the best case). This improvement is made possible by a sharper dimension-free anti-concentration inequality for Gaussian process on a compact metric space. In addition, we prove two non-uniform variants of the high-dimensional CLT based on the large deviation and non-uniform CLT results for random variables in a Banach space by Bentkus, Rackauskas, and Paulauskas. We apply our results in the context of post-selection inference in linear regression and of empirical processes.