High-dimensional CLT: improvements, non-uniform extensions and large deviations. (English) Zbl 1475.60054

Summary: 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. V. Chernozhukov et al. [Ann. Probab. 45, No. 4, 2309–2352 (2017; Zbl 1377.60040)] 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 \(\log^7p=o(n)\). We improve upon their result, for hyperrectangles, which only requires \(\log^4p=o(n)\) (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.


60F05 Central limit and other weak theorems
60F10 Large deviations
60B05 Probability measures on topological spaces
60G15 Gaussian processes


Zbl 1377.60040
Full Text: DOI arXiv Euclid


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