Let $X$ be a real-valued random variable with distribution function $F$. Set $X_1,\dots, X_m$ to be independent copies of $X$ and let $F_m$ be the corresponding empirical distribution function. We show that there are absolute constants $c_0$ and $c_1$ such that if $\Delta \geq c_0\frac{\log\log m}{m}$, then with probability at least $1-2\exp(-c_1\Delta m)$, for every $t\in\mathbb{R}$ that satisfies $F(t)\in[\Delta,1-\Delta]$, \[ |F_m(t) - F(t) | \leq \sqrt{\Delta \min\{F(t),1-F(t)\} } .\] Moreover, this estimate is optimal up to the multiplicative constants $c_0$ and $c_1$.

翻译：设 $X$ 为具有分布函数 $F$ 的实值随机变量。令 $X_1,\dots, X_m$ 为 $X$ 的独立副本，且 $F_m$ 为相应的经验分布函数。我们证明存在绝对常数 $c_0$ 和 $c_1$，使得若 $\Delta \geq c_0\frac{\log\log m}{m}$，则概率至少为 $1-2\exp(-c_1\Delta m)$ 时，对任意满足 $F(t)\in[\Delta,1-\Delta]$ 的 $t\in\mathbb{R}$，有 \[ |F_m(t) - F(t) | \leq \sqrt{\Delta \min\{F(t),1-F(t)\} } .\] 此外，该估计在乘法常数 $c_0$ 和 $c_1$ 意义下是最优的。