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
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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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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