Independance conditionnelle et uniformite pour les lois fortes des grands nombres dans les espaces de Banach. (French) Zbl 0349.60017

Summary: A. Beck has given an “uniform” strong law of large numbers for families of mutually symmetric and uniformly essentially bounded sequences of centered random variables, with values in \((k, \epsilon)\)-\(B\)-convex spaces. We show that, without any limitation on the Banach spaces, the technique used by A. Beck allows to replace, in strong law of large numbers making use of conditions bearing on essential bounds, the hypothesis of independence by an hypothesis called conditional-independence-and-centering, which is weaker than both hypothesis of independence and of mutual symmetry; moreover, in several cases, one gets “uniform” strong laws of large numbers (for families of conditionally-independent-and-centered sequences). The results we get are compared with recent results of G. Pisier, obtained with “type \(p\) spaces” techniques.


60F05 Central limit and other weak theorems
60A05 Axioms; other general questions in probability
Full Text: DOI


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