On a variance dependent Dvoretzky-Kiefer-Wolfowitz inequality

Abstract

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