= Donsker theorem for empirical distribution functions
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{title2=$\sqrt n(F_n-F)\rightsquigarrow\mathbb G_F$}
For an independent sample with <cumulative distribution function> $F$, the <empirical distribution process> converges in the <supremum norm> on $\ell^\infty(\mathbb R)$ to the <F-Brownian bridge>. The convergence uses <outer expectation> and inner expectation when finite-sample Borel measurability fails. It is a full function-space theorem, stronger than the <Finite-dimensional Brownian bridge limit of an empirical distribution process>. No continuity of $F$ is required. The <continuous mapping theorem> gives convergence of the supremum statistic; for continuous $F$, its range contains $(0,1)$ and the bridge vanishes at zero and one, giving the <Kolmogorov-Smirnov theorem>. The function-space statement is stated here as a theorem; its tightness assertion is an additional ingredient beyond the finite-dimensional covariance calculation.
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