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Sur une inegalite de H.P. Rosenthal et le theoreme limite central dans les espaces de Banach

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Abstract

This paper studies an inequality of H. P. Rosenthal for vector valued random variables, its relations with some geometric properties of Banach spaces and its applications to the study of the central limit theorem in Banach spaces.

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Références

  1. A. de Acosta, A. Araujo and E. Giné,On Poisson measures, Gaussian measures and the central limit theorem in Banach spaces, inAdvances in Probability, Vol. 4, Dekker, New York, 1978, pp. 1–68.

    Google Scholar 

  2. T. W. Anderson,The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities, Proc. Am. Math. Soc.6 (1955), 170–176.

    Article  MATH  Google Scholar 

  3. A. Araujo and E. Giné,The Central Limit Theorem for Real and Banach valued Random Variables, Wiley, New York, 1980.

    MATH  Google Scholar 

  4. S. A. Chobanjan and V. I. Tarieladze,Gaussian characterization of certain Banach spaces, J. Multivar. Anal.7 (1977), 183–203.

    Article  MATH  MathSciNet  Google Scholar 

  5. S. A. Chobanjan and V. I. Tarieladze,A counterexample concerning CLT in Banach spaces, inProbability Theory on Vector Spaces, Poland 1977, Lecture Notes in Math.656, Springer, Berlin 1978, pp. 25–30.

    Google Scholar 

  6. X. Fernique,Intégrabilité des vecteurs gaussiens, C. R. Acad. Sci. Paris, Série A,270 (1970), 1698–1699.

    MATH  MathSciNet  Google Scholar 

  7. E. Giné, V. Mandrekar and J. Zinn,On sums of independent random variables with values in L p, 2≦p<∞, inProbability in Banach Spaces II, Oberwolfach 1978, Lecture Notes in Math.709, Springer, Berlin, 1979, pp. 111–124.

    Google Scholar 

  8. E. Giné and J. Zinn,Central limit theorems and weak laws of large numbers in certain Banach spaces, Z. Wahrscheinlichkeitstheor. Verw. Geb.62 (1983), 323–354.

    Article  MATH  Google Scholar 

  9. B. Heinkel,On the law of large numbers in 2-uniformly smooth Banach spaces, Ann. Probab.12 (1984), 851–857.

    MATH  MathSciNet  Google Scholar 

  10. J. Hoffmann-Jørgensen,Sums of independent Banach space valued random variables, Studia Math.52 (1974), 159–186.

    MathSciNet  Google Scholar 

  11. J. Hoffmann-Jørgensen and G. Pisier,The law of large numbers and the central limit theorem in Banach spaces, Ann. Probab.4 (1976), 587–599.

    MATH  Google Scholar 

  12. N. C. Jain,Central limit theorem and related questions in Banach spaces, Proc. Symp. in Pure Math. XXXI, Am. Math. Soc., Providence, 1977, pp. 55–65.

  13. J. P. Kahane,Some Random Series of Functions, Heath, Lexington, 1968.

    MATH  Google Scholar 

  14. J. L. Krivine,Théorèmes de factorisation dans les espaces réticulés, Séminaire Maurey-Schwartz 1973–74, exposés XXII et XXIII, Ecole Polytechnique, Paris, 1974.

    Google Scholar 

  15. J. Kuelbs and J. Zin,Some stability results for vector valued random variables, Ann. Probab.7 (1979), 75–84.

    MATH  MathSciNet  Google Scholar 

  16. H. J. Landau and L. Shepp,On the supremum of a Gaussian process, Sankhya, Ser. A32 (1971), 369–378.

    MathSciNet  Google Scholar 

  17. L. Le Cam,Remarques sur le théorème limite central dans les espaces localement convexes, inLes probabilités sur les structures algébriques, Colloq. C. N. R. S., Paris, 1970, pp. 233–249.

  18. M. Ledoux, Théorème limite central dans les espaces lp(B) (1≦p<∞), Ann. Inst. H. Poincaré19 (1983), 393–411.

    MathSciNet  Google Scholar 

  19. J. Lindenstrauss and L. Tzafriri,Classical Banach Spaces II, Springer, Berlin, 1979.

    MATH  Google Scholar 

  20. B. Maurey,Type et cotype dans les espaces munis de structures locales inconditionnelles, Séminaire Maurey-Schwartz 1973–74, exposés XXIV et XXV, Ecole Polytechnique, Paris, 1974.

    Google Scholar 

  21. B. Maurey and G. Pisier,Séries de variables aléatoires vectorielles indépendantes et propriétés géométriques des espaces de Banach, Studia Math.58 (1976), 45–90.

    MATH  MathSciNet  Google Scholar 

  22. G. Pisier,“Type” des espaces normés, Séminaire Maurey-Schwartz 1973–74, exposé III, Ecole Polytechnique, Paris, 1974.

    Google Scholar 

  23. G. Pisier,Une propriété du type p-stable, Séminaire Maurey-Schwartz 1973–74, exposé VIII, Ecole Polytechnique, Paris, 1974.

    Google Scholar 

  24. G. Pisier,Le théorème de la limite centrale et la loi du logarithme itéré dans les espaces de Banach, Séminaire Maurey-Schwartz 1975–76, exposés III et IV, Ecole Polytechnique, Paris, 1976.

    Google Scholar 

  25. G. Pisier,On the dimension of the l np -subspaces of Banach spaces, for 1≦p<2, Trans. Am. Math. Soc.276 (1983), 201–211.

    Article  MATH  MathSciNet  Google Scholar 

  26. G. Pisier and J. Zinn,On the limit theorems for random variables with values in the spaces L p (2≦p<∞), Z. Wahrscheinlichkeitstheor. Verw. Geb.41 (1977), 289–304.

    Article  MathSciNet  Google Scholar 

  27. H. P. Rosenthal,On the subspaces of L p (p>2) spanned by sequences of independent random variables, Isr. J. Math.8 (1970), 273–303.

    MATH  Google Scholar 

  28. H. P. Rosenthal,On the span in L p of sequences of independent random variables (II), Proc. 6th Berkeley Symp. on Prob. and Stat., Vol. II, Berkeley, Calif., 1971, pp. 149–167.

    Google Scholar 

  29. J. Rosińki,Remarks on Banach spaces of stable type, Prob. Math. Stat.1 (1980), 67–71.

    Google Scholar 

  30. W. A. Woyczyński,On Marcinkiewicz-Zygmund laws of large numbers in Banach spaces and related rates of convergence, Prob. Math. Stat.1 (1980), 117–131.

    Google Scholar 

  31. J. Zinn,Inequalities in Banach spaces with applications to probabilistic limit theorems: a survey, inProbability in Banach Spaces III, Medford 1980, Lecture Notes in Math.860, Springer, Berlin, 1981, pp. 324–329.

    Chapter  Google Scholar 

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  1. Département de Mathématique, Université Louis Pasteur, 7, rue René Descartes, F-67084, Strasbourg Cédex, France

    Michel Ledoux

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  1. Michel Ledoux
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Ledoux, M. Sur une inegalite de H.P. Rosenthal et le theoreme limite central dans les espaces de Banach. Israel J. Math. 50, 290–318 (1985). https://doi.org/10.1007/BF02759762

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  • DOI: https://doi.org/10.1007/BF02759762

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