Special Invited Paper

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


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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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • R. M. Dudley
    • 1
  1. 1.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA

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