  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 $c_1$.
