Abstract

We derive novel anti-concentration bounds for the difference between the maximal values of two Gaussian random vectors across various settings. Our bounds are dimension-free, scaling with the dimension of the Gaussian vectors only through the smaller expected maximum of the Gaussian subvectors. In addition, our bounds hold under the degenerate covariance structures, which previous results do not cover. In addition, we show that our conditions are sharp under the homogeneous component-wise variance setting, while we only impose some mild assumptions on the covariance structures under the heterogeneous variance setting. We apply the new anticoncentration bounds to derive the central limit theorem for the maximizers of discrete empirical processes. Finally, we back up our theoretical findings with comprehensive numerical studies.

About the speaker

Shen Shuting is an Assistant Professor of Statistics & Data Science at the National University of Singapore. Before joining NUS, she was a postdoctoral fellow at the Fuqua School of Business and the Department of Biostatistics & Bioinformatics at Duke University, jointly supervised by Dr. Alexandre Belloni and Dr. Ethan X. Fang. Prior to her postdoctoral position, she obtained her PhD in Biostatistics from Harvard University in 2023, where she was jointly supervised by Dr. Xihong Lin and Dr. Junwei Lu. She earned a B.A. and a B.S. in Mathematics (dual) from Peking University in 2018. Her research interests primarily include large-scale inference, combinatorial inference, choice model asymptotics, operations research theories, applied probability, and distributed computing.