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.

翻译：我们推导了可分离中心高斯过程上确界与高斯随机场顺序统计量的集中函数的匹配下界和上界（精确至常数因子）。这些界表明，可分离中心高斯过程 $\{X_u : u \in U\}$ 的上确界具有与均值为零、方差为 $\mathrm{Var}(\sup_{u \in U} X_u)$ 的单一高斯随机变量相同的反集中性质。为将这些结果应用于高维统计问题，必须理解当指标集 $U$ 的维度或度量熵增大时 $\mathrm{Var}(\sup_{u \in U} X_u)$ 的渐近行为。因此，我们还推导了该量的下界和上界。