2018/06/15 by Arun Kumar Kuchibhotla, Kuchibhotla, Arun Kumar, Somabha Mukherjee +3
Mathematics · #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Statistical Methods and Inference #Statistics Theory (math.ST) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1806.06153
openalex publication_date 2018/06/15 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
Central limit theorems (CLTs) for high-dimensional random vectors with\ndimension possibly growing with the sample size have received a lot of\nattention in the recent times. Chernozhukov et al. (2017) proved a\nBerry--Esseen type result for high-dimensional averages for the class of\nhyperrectangles and they proved that the rate of convergence can be upper\nbounded by n-1/6 upto a polynomial factor of \log p (where n\nrepresents the sample size and p denotes the dimension). Convergence to zero\nof the bound requires \log7p = o(n). We improve upon their result which only\nrequires \log4p = o(n) (in the best case). This improvement is made possible\nby a sharper dimension-free anti-concentration inequality for Gaussian process\non a compact metric space. In addition, we prove two non-uniform variants of\nthe high-dimensional CLT based on the large deviation and non-uniform CLT\nresults for random variables in a Banach space by Bentkus, Ra v ckauskas, and\nPaulauskas. We apply our results in the context of post-selection inference in\nlinear regression and of empirical processes.\n