Anti-Concentration Inequalities for the Difference of Maxima of Gaussian Random Vectors

TL;DR

This paper establishes new anti-concentration bounds for the difference of maxima of Gaussian vectors, which are dimension-free, applicable to degenerate covariances, and have practical implications for empirical process maximizers.

Contribution

It introduces novel, sharp anti-concentration bounds for Gaussian maxima differences that extend to degenerate covariance structures and are dimension-free.

Findings

Bounds are dimension-free and depend only on the smaller expected maximum.

Applicable to degenerate covariance structures, broadening previous results.

Used to derive a CLT for maximizers of empirical processes.

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 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.