Zusammenfassung
Although there is an extensive literature on the maxima of Gaussian
processes, there are relatively few non-asymptotic bounds on their lower-tail
probabilities. In the context of a finite index set, this paper offers such a
bound, while also allowing for many types of dependence. Specifically, let
$(X_1,\dots,X_n)$ be a centered Gaussian vector, with standardized entries,
whose correlation matrix $R$ satisfies $\max_ij R_ij\rho_0$ for
some constant $\rho_0(0,1)$. Then, for any $\epsilon_0ın
(0,1-\rho_0)$, we establish an upper bound on the probability
$P(\max_1in X_i\epsilon_02łog(n))$ that is a
function of $\rho_0, \, \epsilon_0,$ and $n$. Furthermore, we show the bound is
sharp, in the sense that it is attained up to a constant, for each $\rho_0$ and
$\epsilon_0$.
Nutzer