 Dvoretzky–Kiefer–Wolfowitz inequality

For independent and identically distributed real random variables with cumulative distribution function $F$, the Dvoretzky-Kiefer-Wolfowitz inequality gives, for every $n\ge 1$ and $\epsilon >0$,The supremum is the Kolmogorov distance: a single event controls all thresholds simultaneously. The bound holds for discrete, continuous, and mixed probability distributions.^[ref]

For one fixed threshold, Hoeffding's inequality gives the same expression as an upper bound on $\mathbb{P}(|F_n(x)-F(x)|>\epsilon )$. 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.

The Dvoretzky–Kiefer–Wolfowitz (DKW) inequality is a result in probability theory concerning the convergence of the empirical distribution function to the true cumulative distribution function. Specifically, it provides a bound on the probability that the empirical distribution function deviates from the true distribution function by more than a certain amount.