Under the same hypotheses as the Donsker–Kolmogorov–Doob theorem, as $n \to \infty$,
\begin{align*}
\sqrt{n}\,\|F_n - F\|_\infty \xrightarrow{d} \|G_F\|_\infty = \sup_{t \in [0,1]} |B_t|,
\end{align*}
where $(B_t)_{t \in [0,1]}$ is a standard Brownian bridge.