Donsker classes and random geometry
Let \({\mathcal F}\) be a class of square integrable functions. The empirical process indexed by \({\mathcal F}\) is investigated. The paper presents a refinement of the results of \textit{E. Giné} and \textit{J. Zinn}, Ann. Probab. 12, 929-989 (1984; Zbl 0553.60037), concerning the CLT under random entropy conditions. Necessary and sufficient conditions are found. They mean that the trace of \({\mathcal F}\) on a random sample is a small (for the \(\ell^ 1\) norm) perturbation of a set which is nice for the \(\ell^ 2\) norm.
scientific article; zbMATH DE number 4038916