On a variance dependent Dvoretzky-Kiefer-Wolfowitz inequality

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 $\mathrm{\Delta }\ge c_0\frac{\log \log m}{m}$, then with probability at least $1-2\exp (-c_1\mathrm{\Delta }m)$, for every $t\in \mathbb{R}$ that satisfies $F(t)\in [\mathrm{\Delta },1-\mathrm{\Delta }]$, $$\mathrm{\mid }{F}_m(t)-F(t)\mathrm{\mid }\le \sqrt{\mathrm{\Delta }\min \{F(t),1-F(t)\}}\mathrm{.}$$ Moreover, this estimate is optimal up to the multiplicative constants $c_0$ and $c_1$.