Comparison and anti-concentration bounds for maxima of Gaussian random vectors

TL;DR

This paper provides explicit comparison inequalities and a dimension-free anti-concentration bound for the maxima of Gaussian vectors, with applications to high-dimensional statistical inference.

Contribution

It introduces new comparison and anti-concentration bounds for Gaussian maxima without covariance restrictions, extending previous inequalities.

Findings

Established a dimension-free anti-concentration inequality for Gaussian maxima.

Derived bounds on the Kolmogorov distance between Gaussian maxima distributions.

Applied results to high-dimensional central limit theorems for sums of random vectors.

Abstract

Slepian and Sudakov-Fernique type inequalities, which compare expectations of maxima of Gaussian random vectors under certain restrictions on the covariance matrices, play an important role in probability theory, especially in empirical process and extreme value theories. Here we give explicit comparisons of expectations of smooth functions and distribution functions of maxima of Gaussian random vectors without any restriction on the covariance matrices. We also establish an anti-concentration inequality for the maximum of a Gaussian random vector, which derives a useful upper bound on the L\'{e}vy concentration function for the Gaussian maximum. The bound is dimension-free and applies to vectors with arbitrary covariance matrices. This anti-concentration inequality plays a crucial role in establishing bounds on the Kolmogorov distance between maxima of Gaussian random vectors. These results have immediate applications in mathematical statistics. As an example of application, we establish a conditional multiplier central limit theorem for maxima of sums of independent random vectors where the dimension of the vectors is possibly much larger than the sample size.