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Israel Journal of Mathematics

, Volume 50, Issue 4, pp 290–318 | Cite as

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

  • Michel Ledoux
Article

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. 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. 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.zbMATHCrossRefGoogle Scholar
  3. 3.
    A. Araujo and E. Giné,The Central Limit Theorem for Real and Banach valued Random Variables, Wiley, New York, 1980.zbMATHGoogle Scholar
  4. 4.
    S. A. Chobanjan and V. I. Tarieladze,Gaussian characterization of certain Banach spaces, J. Multivar. Anal.7 (1977), 183–203.zbMATHCrossRefMathSciNetGoogle Scholar
  5. 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. 6.
    X. Fernique,Intégrabilité des vecteurs gaussiens, C. R. Acad. Sci. Paris, Série A,270 (1970), 1698–1699.zbMATHMathSciNetGoogle Scholar
  7. 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. 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.zbMATHCrossRefGoogle Scholar
  9. 9.
    B. Heinkel,On the law of large numbers in 2-uniformly smooth Banach spaces, Ann. Probab.12 (1984), 851–857.zbMATHMathSciNetGoogle Scholar
  10. 10.
    J. Hoffmann-Jørgensen,Sums of independent Banach space valued random variables, Studia Math.52 (1974), 159–186.MathSciNetGoogle Scholar
  11. 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.zbMATHGoogle Scholar
  12. 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.Google Scholar
  13. 13.
    J. P. Kahane,Some Random Series of Functions, Heath, Lexington, 1968.zbMATHGoogle Scholar
  14. 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. 15.
    J. Kuelbs and J. Zin,Some stability results for vector valued random variables, Ann. Probab.7 (1979), 75–84.zbMATHMathSciNetGoogle Scholar
  16. 16.
    H. J. Landau and L. Shepp,On the supremum of a Gaussian process, Sankhya, Ser. A32 (1971), 369–378.MathSciNetGoogle Scholar
  17. 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.Google Scholar
  18. 18.
    M. Ledoux, Théorème limite central dans les espaces lp(B) (1≦p<∞), Ann. Inst. H. Poincaré19 (1983), 393–411.MathSciNetGoogle Scholar
  19. 19.
    J. Lindenstrauss and L. Tzafriri,Classical Banach Spaces II, Springer, Berlin, 1979.zbMATHGoogle Scholar
  20. 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. 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.zbMATHMathSciNetGoogle Scholar
  22. 22.
    G. Pisier,“Type” des espaces normés, Séminaire Maurey-Schwartz 1973–74, exposé III, Ecole Polytechnique, Paris, 1974.Google Scholar
  23. 23.
    G. Pisier,Une propriété du type p-stable, Séminaire Maurey-Schwartz 1973–74, exposé VIII, Ecole Polytechnique, Paris, 1974.Google Scholar
  24. 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. 25.
    G. Pisier,On the dimension of the l pn-subspaces of Banach spaces, for 1≦p<2, Trans. Am. Math. Soc.276 (1983), 201–211.zbMATHCrossRefMathSciNetGoogle Scholar
  26. 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.CrossRefMathSciNetGoogle Scholar
  27. 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.zbMATHGoogle Scholar
  28. 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. 29.
    J. Rosińki,Remarks on Banach spaces of stable type, Prob. Math. Stat.1 (1980), 67–71.Google Scholar
  30. 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. 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.CrossRefGoogle Scholar

Copyright information

© The Weizmann Science Press of Israel 1985

Authors and Affiliations

  • Michel Ledoux
    • 1
  1. 1.Département de MathématiqueUniversité Louis PasteurStrasbourg CédexFrance

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