Anti-concentration of Suprema of Gaussian Processes and Gaussian Order Statistics

We derive, up to a constant factor, matching lower and upper bounds on the concentration functions of suprema of separable centered Gaussian processes and order statistics of Gaussian random fields. These bounds reveal that suprema of separable centered Gaussian processes $\{X_u : u \in U\}$ exhibit the same anti-concentration properties as a single Gaussian random variable with mean zero and variance $\mathrm{Var}(\sup_{u \in U} X_u)$. To apply these results to high-dimensional statistical problems, it is therefore essential to understand the asymptotic behavior of $\mathrm{Var}(\sup_{u \in U} X_u)$ as the dimension or metric entropy of the index set $U$ increases. Consequently, we also derive lower and upper bounds on this quantity.