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

Abstract

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 $({\mathrm{\xi }}_1,\dots ,{\mathrm{\xi }}_N)$ be a centered Gaussian vector with standardized entries, whose correlation matrix R satisfies ${max}_{i\ne j}{R}_{ij}\le {\mathit{\rho }}_0$ for some constant ${\mathit{\rho }}_0\in (0,1)$. Then, for any ${\mathit{ϵ}}_0\in (0,\sqrt{1-{\mathit{\rho }}_0})$, we establish an upper bound on the probability $\mathbb{P}({max}_{1\le j\le N}{\mathrm{\xi }}_j\le {\mathit{ϵ}}_0\sqrt{2log(N)})$ in terms of $({\mathit{\rho }}_0,{\mathit{ϵ}}_0,N)$. 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”.