# Special Invited Paper

Central Limit Theorems for Empirical Measures
• R. M. Dudley
Chapter
Part of the Selected Works in Probability and Statistics book series (SWPS)

## Abstract

The statistics used in Kolmogorov-Smirnov test are suprema of normalized empirical measures $${n^\frac{1}{2}\left(P_n-P\right)}$$ or $$\left(mn\right)^\frac{1}{2}{\left({m+n}\right)}^{-\frac{1}{2}}\left({P_m}-{Q_n}\right)$$ over a class $$\mathcal{C}$$ of sets, namely the interval $$]-{\infty},a],a \in {\mathbb{R}}.$$ Donsker (1952) Showed here that $${n^\frac{1}{2}\left(P_n-P\right)}$$ converges in law, in the spacel $$l^{\infty}\left({\mathcal{C}}\right)$$ of all bounded functions products of interval parallel to the axes in $$\mathbb{R}^k$$ (Dudley (1966), (1967a)). Since $$\ell^\infty\mathcal{C}$$ in the supremum norm is nonseparable, some measurablity problems (overlooked by Dansker) had to be trated. Recently Révész (1976) proved an iterated logarithm law for a much more general class of sets
$$\bigcap_{1\leq i\leq k}\left\{x:{f}_i\left\{\left(x_j:\neq i\right)\right\}< {x}_i < {g}_i\left(\left\{ {x_j:j\neq i}\right\}\right)\right\}$$
where $${f}_i$$ and g i have a fixed bound on their partial derivatives of orders $$\leq k,$$ and $$\mathbf{P}$$ is the uniform measure on the unit cube. This paper will consider extensions of Donsker’s theorem to suitable classes of sets in general probability spaces.

## Key words and phrases

Central limit theorems empirical measures Donsker classes Effros Borel structure metric entropy with inclusion two-sample case Vapnik-Červonenkis classes

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