An application of a martingale inequality of dubins and freedman to the law of large numbers in Banach spaces

  • Bernard Heinkel
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1193)


In a real, separable, p-uniformly smooth Banach space the law of large numbers in the Prohorov setting is studied by a method depending on a result of Dubins and Freedman which compares the distribution of a real valued martingale with the one of the associated conditional variances. Some laws of large numbers of Kolmogorov-Brunk type are also given.


Banach Space Conditional Variance Iterate Logarithm Smooth Banach Space Smooth Space 
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  1. [1]
    DE ACOSTA, A.: Inequalities for B-valued random vectors with applications to the strong law of large numbers. Ann. Prob. 9 (1981), 157–161.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    BRUNEL, A. et SUCHESTON, L.: Sur les amarts faibles à valeurs vectorielles. C.R. Acad. Sc. Paris 282, Sér. A (1976), 1011–1014MathSciNetzbMATHGoogle Scholar
  3. [3]
    DUBINS, L.E. and FREEDMAN, D.A.: A sharper form of the Borel-Cantelli lemma and the strong law. Ann. Math. Stat. 36 (1965), 800–807MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    EGGHE, L.: Stopping time techniques for analysts and probabilists. Cambridge University Press-Cambridge 1984CrossRefzbMATHGoogle Scholar
  5. [5]
    ENFLO, P.: Banach spaces which can be given an equivalent uniformly convex norm. Israel J. of Math. 13 (1972), 281–288MathSciNetCrossRefGoogle Scholar
  6. [6]
    GODBOLE, A.: Strong laws of large numbers and laws of the iterated logarithm in Banach spaces. PHD, Michigan State University 1984Google Scholar
  7. [7]
    HEINKEL, B.: Relation entre théorème central-limite et loi du logarithme itéré dans les espaces de Banach. Z. Wahrscheinlichkeitstheorie verw. Gebiete 49 (1979), 211–220MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    HEINKEL, B.: Sur la loi du logarithme itéré dans les espaces réflexifs. Séminaire de Probabilités 16-1980/81-Lecture Notes in Math 920, 602–608Google Scholar
  9. [9]
    HEINKEL, B.: On the law of large numbers in 2-uniformly smooth Banach spaces. Ann. Prob. 12 (1984), 851–857MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    HEINKEL, B.: The non i.i.d. strong law of large numbers in 2-uniformly smooth Banach spaces. Probability Theory on Vector Spaces III-Lublin 1983 Lecture Notes in Math 1080, 90–118.Google Scholar
  11. [11]
    HEINKEL, B.: Une extension de la loi des grands nombres de Prohorov. Z. Wahrscheinlichkeitstheorie verw. Gebiete 67 (1984), 349–362MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    HEINKEL, B.: On Brunk's law of large numbers in some type 2 spaces. to appear in "Probability in Banach spaces 5"-Medford 1984-, Lecture Notes in MathGoogle Scholar
  13. [13]
    HOFFMANN-JØRGENSEN, J.: Sums of independent Banach space valued random variables. Studia Math 52 (1974), 159–186MathSciNetzbMATHGoogle Scholar
  14. [14]
    HOFFMANN-JØRGENSEN, J.: Probability in Banach space. Ecole d'été de Probabilités de St Flour 6-1976-Lecture Notes in Math 598, 1–186Google Scholar
  15. [15]
    HOFFMANN-JØRGENSEN, J. and PISIER, G.: The law of large numbers and the central limit theorem in Banach spaces. Ann. Prob. 4 (1976), 587–599MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    KUELBS, J.: Kolmogorov law of the iterated logarithm for Banach space valued random variables. Ill. J. of Math. 21 (1977), 784–800MathSciNetzbMATHGoogle Scholar
  17. [17]
    KUELBS, J. and ZINN, J.: Some stability results for vector valued random variables. Ann. Prob. 7 (1979), 75–84MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    LEDOUX, M.: Sur une inégalité de H.P. Rosenthal et le théorème limite central dans les espaces de Banach. Israel J. of Math. 50 (1985), 290–318MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    LEDOUX, M.: La loi du logarithme itéré dans les espaces de Banach uniformément convexes. C. R. Acad. Sc. Paris 300, Sér. I, no 17 (1985), 613–616MathSciNetzbMATHGoogle Scholar
  20. [20]
    STOUT, W.F.: Almost sure convergence. Academic Press, New York 1974zbMATHGoogle Scholar
  21. [21]
    YURINSKII, V.V.: Exponential bounds for large deviations. Theor. Prob. Appl. 19 (1974), 154–155MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • Bernard Heinkel
    • 1
  1. 1.Département de MathématiqueStrasbourg CédexFrance

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