A sharp lower-tail bound for Gaussian maxima with application to bootstrap methods in high dimensions

Although there is an extensive literature on the maxima of Gaussian processes, there are relatively few non-asymptotic bounds on their lower-tail probabilities. The aim of this paper is to develop such a bound, while also allowing for many types of dependence. Let (xi(1),..., xi(N)) be a centered Gaussian vector with standardized entries, whose correlation matrix R satisfies max(i not equal j) R-ij <= rho 0 for some constant rho 0 is an element of (0, 1). Then, for any epsilon(0). (0, root 1 -rho(0)), we establish an upper bound on the probability P(max1 <= j <= N xi(j) <= epsilon(0)root 2 log(N)) in terms of (rho 0, epsilon 0, N). The bound is also sharp, in the sense that it is attained up to a constant, independent of N. Next, we apply this result in the context of high-dimensional statistics, where we simplify and weaken conditions that have recently been used to establish near-parametric rates of bootstrap approximation. Lastly, an interesting aspect of this application is that it makes use of recent refinements of Bourgain and Tzafriri's "restricted invertibility principle".