This paper deals with the Gaussian and bootstrap approximations to the distribution of the max statistic in high dimensions. This statistic takes the form of the maximum over components of the sum of independent random vectors and its distribution plays a key role in many high-dimensional estimation and testing problems. Using a novel iterative randomized Lindeberg method, the paper derives new bounds for the distributional approximation errors. These new bounds substantially improve upon existing ones and simultaneously allow for a larger class of bootstrap methods.
K. Kato is suport by the NSF Grant DMS-1952306 and DMS-2014636.
We are grateful to Tim Armstrong, Matias Cattaneo, Xiaohong Chen and Tengyuan Liang for helpful discussions. We also thank seminar participants at the University of Pennsylvania and Yale University.
"Improved central limit theorem and bootstrap approximations in high dimensions." Ann. Statist. 50 (5) 2562 - 2586, October 2022. https://doi.org/10.1214/22-AOS2193