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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.

MSC:
60F05 Central limit and other weak theorems
60A05 Axioms; other general questions in probability
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[1] Beck, A.: On the strong law of large numbers. Ergodic theory, Proc. Inst. Sympos. New Orleans, F.B. Wright, ed. 21-53 (1961)
[2] Beck, A.: Conditional independence. Z. Wahrscheinlichkeitstheorie verw. Gebiete, 33, 253-267 (1976) · Zbl 0363.60030
[3] Beck, A., Giesy, D.P.: P-uniform convergence and a vector valued strong law of large numbers. Trans. Amer. Math. Soc. 147, 541-559 (1970) · Zbl 0198.51006
[4] Chatterji, S.D.: Comments on the Martingale Convergence theorem. Symposium on probability methods in Analysis, Lecture notes in Mathematics, 31. Berlin-Heidelberg-New York: Springer 1967 · Zbl 0178.19902
[5] Day, M.M.: Normed linear spaces (3rd edition). Berlin-Heidelberg-New York: Springer 1973 · Zbl 0268.46013
[6] Giesy, D.P.: On a convexity condition in normed linear spaces. Trans. Amer. Math. Soc. 125, 114-146 (1966) · Zbl 0183.13204
[7] Giesy, D.P.: Strong law of large numbers for independent sequences of Banach space valued random variables. Probability in Banach spaces, Oberwolfach 1975. Lecture notes in Mathematics 526, 89-99. Berlin-Heidelberg-New York: Springer 1976
[8] Hoffmann-Jørgensen, J., Pisier, G.: The law of large numbers and the central limit theorem in Banach spaces. Ann. Probability 4, 587-599 (1976) · Zbl 0368.60022
[9] Loeve, M.: Probability theory. New York: Van Nostrand 1960 · Zbl 0108.14202
[10] Parthasarathy, K.R.: Probability measures on metric spaces. New York-London: Academic Press 1967 · Zbl 0153.19101
[11] Pisier, G.: B-convexity, super-reflexivity and the ?three space problem?. Proceedings of the Conference on random series, convex sets and the geometry of Banach spaces, Aarhus Oct. 1974, Various publication series, Aarhus University (1975)
[12] Pisier, G.: Communication privée à A. Beck, citée dans A. Beck, Cancellation in Banach spaces, Probability in Banach spaces, Oberwolfach 1975. Lecture notes in Mathematics 526, 13-20. Berlin-Heidelberg-New York: Springer 1976
[13] Pisier, G.: Communication privée à E. Ménard et J. P. Raoult. Ecole Polytechnique, Palaiseau, France. Le 30 nov. 1976
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