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For an empirical distribution function based on an independent sample from a continuous distribution $F$,
$$
\sqrt n\sup_x|\widehat F_n(x)-F(x)|
\xrightarrow{d}\sup_{0\leq t\leq1}|B(t)|,
$$
where $B$ is a <Brownian bridge>.

This limit calibrates the <Kolmogorov-Smirnov test>; the test compares an empirical distribution with its null distribution using the supremum discrepancy.

Kolmogorov-Smirnov theorem

Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 31 / 1 / Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 29J / a / Solution

Kolmogorov-Smirnov theorem

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Kolmogorov-Smirnov theorem

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