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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

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.

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.)
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