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Special Invited Paper

Central Limit Theorems for Empirical Measures

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Book cover Selected Works of R.M. Dudley

Part of the book series: Selected Works in Probability and Statistics ((SWPS))

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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.

Received June 15, 1977; revised

This reserach was partially supported by National Science Foundation Grant MCS76-07211 A01.

AMS 1970 subject classification. Primary 60F05; Secondary 60B10, 60G17, 28A05, 28A40.

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Authors and Affiliations

  1. Department of Mathematics, Massachusetts Institute of Technology, Room 2-245, Cambridge, Massachusetts, 02139, USA

    R. M. Dudley

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  1. R. M. Dudley
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Editors and Affiliations

  1. , Mathematics, University of Connecticut, Auditorium Road 196, Storrs, 06269-3009, USA

    Evarist Giné

  2. , Mathematics, Georgia Institute of Technology, Atlanta, 30332-0160, USA

    Vladimir Koltchinskii

  3. Inst. Mathematics & Informatics, Akademijos 4, Vilnius, LT-08663, Lithuania

    Rimas Norvaisa

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Dudley, R.M. (2010). Special Invited Paper. In: Giné, E., Koltchinskii, V., Norvaisa, R. (eds) Selected Works of R.M. Dudley. Selected Works in Probability and Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-5821-1_16

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