Let $F_n(x)=\frac{1}{n}\sum^{n}_{i=1} \text{1} _{\{X_{i}\leq x\}}$ be an empirical distribution function
by Dvoretzky–Kiefer–Wolfowitz inequality:
$Pr(\underset{x\in\mathbb{R}}{\text{sup}}|F_n(x)-F(x)|>\epsilon)\leq 2e^{-2n\epsilon^{2}} \quad \forall \epsilon>0$
I am looking for a similar result for entropy estimates for instance: $Pr(|\hat{H}_n-H|>\epsilon)$ to be upper bounded.