A scale-free density bound for Gaussian maxima

Abstract

We derive a scale-free bound on the density of the maximum of a centered Gaussian vector. The basic bound is non-uniform, depends logarithmically on the dimension, and allows any covariance matrix. When the largest marginal variance is separated from zero, it implies that the density of the maximum is uniformly controlled at all quantiles above $2/3$, which is sufficient for many hypothesis testing applications; it yields validity of Gaussian and bootstrap approximations for maxima of high-dimensional sums at test levels $\alpha <1/3$ without further restricting the covariance. Under these same conditions, the argument is extended to show that the maximum absolute value of a Gaussian vector has a uniformly bounded density on the real line. The method also produces new bounds on the variance of the maximum. We discuss implications for high-dimensional correlation testing, time-uniform sequential inference, and non-parametric confidence bands under latent, low-dimensional structure.