Special Invited Paper

摘要: 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 \(\mathcal{C}\) sets, namely the interval \(]-{\infty},a],a \in {\mathbb{R}}.\) Donsker (1952) Showed here that converges law, spacel \(l^{\infty}\left({\mathcal{C}}\right)\) all bounded functions products parallel to axes \(\mathbb{R}^k\) (Dudley (1966), (1967a)). Since \(\ell^\infty\mathcal{C}\) supremum norm is nonseparable, some measurablity problems (overlooked by Dansker) had be trated. Recently Revesz (1976) proved an iterated logarithm law for much more general 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 fixed bound on their partial derivatives orders \(\leq k,\) \(\mathbf{P}\) uniform measure unit cube. This paper will consider extensions Donsker’s theorem suitable classes probability spaces.