Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On the strong law of large numbers in Banach spaces

Author: Anant P. Godbole
Journal: Proc. Amer. Math. Soc. 100 (1987), 543-550
MSC: Primary 60B11; Secondary 46B20, 60B12
MathSciNet review: 891161
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study the relationship between the geometry of a real separable Banach space $ B$ (as manifested in its cotype, type, or logtype) and necessary or sufficient criteria for the validity of the Strong Law of Large Numbers (SLLN) for independent $ B$-valued random variables, formulated in terms of the validity of a (verifiable) SLLN for real-valued random variables. Our results are the best possible of their kind and may be used in situations where the SLLN's of Hoffman-Jørgensen and Pisier, and Kuelbs and Zinn are inconclusive.

References [Enhancements On Off] (What's this?)

  • [1] Anatole Beck, Daniel P. Giesy, and Peter Warren, Recent developments in the theory of strong laws of large numbers for vector-valued random variables, Teor. Verojatnost. i Primenen. 20 (1975), 126–133 (English, with Russian summary). MR 0391231
  • [2] A. P. Godbole, Dissertation, Michigan State University, East Lansing, Michigan, 1984.
  • [3] Jørgen Hoffmann-Jørgensen, Sums of independent Banach space valued random variables, Studia Math. 52 (1974), 159–186. MR 0356155
  • [4] J. Hoffmann-Jørgensen and G. Pisier, The law of large numbers and the central limit theorem in Banach spaces, Ann. Probability 4 (1976), no. 4, 587–599. MR 0423451
  • [5] J. Kuelbs and Joel Zinn, Some stability results for vector valued random variables, Ann. Probab. 7 (1979), no. 1, 75–84. MR 515814
  • [6] S. Kwapień, A theorem on the Rademacher series with vector valued coefficients, Probability in Banach spaces (Proc. First Internat. Conf., Oberwolfach, 1975), Springer, Berlin, 1976, pp. 157–158. Lecture Notes in Math., Vol. 526. MR 0451333
  • [7] Michel Loève, Probability theory, Third edition, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1963. MR 0203748
  • [8] S. V. Nagaev, Necessary and sufficient conditions for the strong law of large numbers, Teor. Verojatnost. i Primenen. 17 (1972), 609–618 (Russian, with English summary). MR 0312548
  • [9] Yu. V. Prokhorov, Some remarks on the strong law of large numbers, Theor. Probability Appl. 4 (1959), 204–208. MR 0121858
  • [10] William F. Stout, Almost sure convergence, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974. Probability and Mathematical Statistics, Vol. 24. MR 0455094
  • [11] N. A. Volodin and S. V. Nagaev, A remark on the strong law of large numbers, Theor. Probab. Appl. 23 (1978), 810-813.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 60B11, 46B20, 60B12

Retrieve articles in all journals with MSC: 60B11, 46B20, 60B12

Additional Information

Keywords: Type $ p$ and cotype $ q$ Banach spaces, Strong Law of Large Numbers
Article copyright: © Copyright 1987 American Mathematical Society