= Confidence band
{title2=$\mathbb P(L(t)\leq f(t)\leq U(t)\text{ for every }t)\geq1-\alpha$}
A confidence band provides simultaneous bounds on an unknown function over its entire index set. Its coverage event is the intersection over all indices, unlike separate pointwise <confidence intervals>. For a continuous <cumulative distribution function>, the <Kolmogorov-Smirnov theorem> gives an asymptotic band $F_n\pm c_\alpha/\sqrt n$, clipped to $[0,1]$, where $c_\alpha$ is a $(1-\alpha)$-quantile of the absolute <Brownian bridge> maximum. The simultaneous event equals $\sqrt n\|F_n-F\|_\infty\leq c_\alpha$, so <convergence in distribution> at this continuity point proves coverage tending to $1-\alpha$.
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