A Sharp Lower-tail Bound for Gaussian Maxima with Application to Bootstrap Methods in High Dimensions

TL;DR

This paper derives a sharp, non-asymptotic lower-tail bound for the maximum of Gaussian vectors with dependence, and applies it to improve bootstrap methods in high-dimensional statistics.

Contribution

It provides a new non-asymptotic lower-tail bound for Gaussian maxima with dependence, and demonstrates its application to enhance bootstrap approximation in high dimensions.

Findings

Established a sharp upper bound on lower-tail probabilities for Gaussian maxima.

Simplified conditions for bootstrap approximation in high-dimensional settings.

Utilized recent refinements of the restricted invertibility principle in statistical analysis.

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 $(\xi_1,\dots,\xi_N)$ be a centered Gaussian vector with standardized entries, whose correlation matrix $R$ satisfies $\max_{i\neq j} R_{ij}\leq \rho_0$ for some constant $\rho_0\in (0,1)$. Then, for any $\epsilon_0\in(0,\sqrt{1-\rho_0})$, we establish an upper bound on the probability $\mathbb{P}(\max_{1\leq j\leq N} \xi_j\leq \epsilon_0\sqrt{2\log(N)})$ 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".