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".
翻译：虽然有关于高斯进程最大值的广泛文献, 但相对而言, 在低尾概率上, 没有多少非参数值。 本文的目的是开发这样的约束值, 同时允许多种依赖性 。 $( xi_ 1,\ dots,\xi_ N) 是一个带有标准条目的高斯矢量中心, 其相关矩阵值$xi_ i\ neq j} Rí ⁇ leq\rho_ 0_ rho_ 0美元, 一些恒定值$\rho_ 0\ in ( 0, 1 美元) 。 然后, 对于任何 eepsilon_ 0\ in ( 0,\\\ rho_ 0} 美元), 本文的目的是为了开发这样的约束值 。 $\\\\\\\\ \\\\ \\\\\\\\\\\\\\\\\\\\\\\\\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
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