The Glivenko-Cantelli problem, ten years later
Loosely speaking, a Glivenko-Cantelli (GC) class \(Z\) consists of measurable uniformly bounded functions on a complete probability space \((\Omega, {\mathcal F}, {\mathbf P})\) s.t., given a sequence of \(\Omega\)-valued independent r.v.'s \(X_k\) of law \({\mathbf P}\), one has \[ \lim_{n\to\infty}{\mathbf E}\sup \Biggl\{\frac{1}{n}\Bigl|\sum_{k\leq n}(f(X_k)-{\mathbf E}f)\Bigr|,\;f\in Z\Biggr\}=0. \] Various results related to characterisation of GC classes are discussed. The author has simplified his previous difficult proofs obtained ten years ago [Ann. Probab. 15, 837-870 (1987; Zbl 0632.60024)]. The accent is made on the cases where involved analysis of measurability problems can be avoided.
scientific article; zbMATH DE number 1000174
scientific article