Bru, Bernard; Heinich, Henri; Lootgieter, Jean-Claude Lévy distances and extensions of the central limit theorem and of the Glivenko-Cantelli theorem. (Distances de Lévy et extensions des théorèmes de la limite centrale et de Glivenko-Cantelli.) (French) Zbl 0786.60006 Publ. Inst. Stat. Univ. Paris 37, No. 3-4, 29-42 (1993). Let \(P_ E\) be the space of laws of random variables in an r.i. Köthe space \(E\). Let \[ d_ E(\mu,\nu)=\inf \bigl\{ \| X-Y \|_ E;\;{\mathfrak L}(X)=\mu,{\mathfrak L} (Y)=\nu \bigr\}. \] For the distance \(d_ E\) the authors investigate the central limit theorem (when \(L^ p \subset E \subset L^ 2\), where \(p \geq 2\) and the embeddings are continuous) and a Glivenko-Cantelli theorem (when \(L^ p \subset E \subset L^ 1\), where \(p<\infty)\). The existence and unicity of the expectation of Doss in Banach space and in \((P_ E,d_ E)\) is examined, too. Reviewer: A.J.Rachkauskas (Vilnius) Cited in 5 Documents MSC: 60B10 Convergence of probability measures 60B12 Limit theorems for vector-valued random variables (infinite-dimensional case) 60B05 Probability measures on topological spaces 46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) Keywords:Köthe space; central limit theorem; Glivenko-Cantelli theorem PDFBibTeX XMLCite \textit{B. Bru} et al., Publ. Inst. Stat. Univ. Paris 37, No. 3--4, 29--42 (1993; Zbl 0786.60006)