Open Access
February 2021 High-dimensional CLT: Improvements, non-uniform extensions and large deviations
Arun Kumar Kuchibhotla, Somabha Mukherjee, Debapratim Banerjee
Bernoulli 27(1): 192-217 (February 2021). DOI: 10.3150/20-BEJ1233


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.


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Arun Kumar Kuchibhotla. Somabha Mukherjee. Debapratim Banerjee. "High-dimensional CLT: Improvements, non-uniform extensions and large deviations." Bernoulli 27 (1) 192 - 217, February 2021.


Received: 1 June 2019; Revised: 1 February 2020; Published: February 2021
First available in Project Euclid: 20 November 2020

zbMATH: 07282847
MathSciNet: MR4177366
Digital Object Identifier: 10.3150/20-BEJ1233

Keywords: anti-concentration , Cramér type large deviation , Empirical processes , nonuniform CLT , Orlicz norms , Post-selection inference

Rights: Copyright © 2021 Bernoulli Society for Mathematical Statistics and Probability

Vol.27 • No. 1 • February 2021
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