= Dvoretzky-Kiefer-Wolfowitz inequality
{c}
{title2=$\mathbb P(\sup_x|F_n(x)-F(x)|>\varepsilon)\leq2e^{-2n\varepsilon^2}$}
{wiki=Dvoretzky–Kiefer–Wolfowitz_inequality}
= DKW inequality
{c}
{synonym}
For <independent and identically distributed> real <random variables> with <cumulative distribution function> $F$, the <Dvoretzky-Kiefer-Wolfowitz inequality> gives, for every $n\geq1$ and $\varepsilon>0$,
$$
\mathbb P\!\left(\sup_{x\in\mathbb R}|F_n(x)-F(x)|>\varepsilon\right)\leq2e^{-2n\varepsilon^2}.
$$
The <supremum> is the <Kolmogorov distance>: a single <event> controls all thresholds simultaneously. The bound holds for discrete, continuous, and mixed <probability distributions>.<All of Nonparametric Statistics by Larry Wasserman>[§2.1, printed p. 14, Theorem 2.4(3), equation (2.5)]{ref}
For one fixed threshold, <Hoeffding's inequality> gives the same expression as an upper bound on $\mathbb P(|F_n(x)-F(x)|>\varepsilon)$. That fixed-threshold result alone does not justify moving the <supremum> inside the <probability>. The <DKW inequality> is the stronger statement needed for a <simultaneous empirical distribution confidence band>.
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