2024/08/23 by Alexandre Belloni, Belloni, Alexandre, Ethan X. Fang +3
Decision Sciences · Mathematics · #FOS: Mathematics #Point processes and geometric inequalities #Probability (math.PR) #Probability and Risk Models #Statistical Methods and Inference #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.2408.13348
openalex publication_date 2024/08/23 · openalex created_date 2024/09/21 · openalex updated_date 2026/07/28
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 anti-concentration 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.