Journal of Theoretical Probability

, Volume 13, Issue 1, pp 85–92 | Cite as

Spitzer's Strong Law of Large Numbers in Nonseparable Banach Spaces

  • Berthold Wittje


It is well known, that for the sums of i.i.d. random variables we have Sn/n → 0 a.s. iff ∑n=1 1/nP(|Sn| > ) < ∞ holds for all ε > 0 (Spitzer's SLLN). The result is also known in separable Banach spaces. It will be shown, that this also holds in nonseparable (= not necessarily separable) Banach spaces without any measurability assumption. In the theory of empirical processes this gives a characterization of Glivenko-Cantelli classes.

strong law of large numbers Glivenko–Cantelli class nonmeasurable function 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Anderson, N. T. (1985a). The calculus of non-measurable functions and sets, Various Publications Series, No. 36, Aarhus.Google Scholar
  2. 2.
    Anderson, N. T. (1985b). The central limit theorem for non-separable valued functions. Zeitschrift für Wahrsch. verw. Geb. 70, 445–455.CrossRefGoogle Scholar
  3. 3.
    Azlarov, T. A., and Volodin, N. A. (1981). Laws of large numbers for identically distributed Banach-space valued random variables. Theor. Prob. Appl. 26, 473–580.MathSciNetzbMATHGoogle Scholar
  4. 4.
    Dvoretzky, A., Kiefer, J., and Wolfowitz, J. (1956). Asymptotic minimax character of the sample distribution functions and of the classical multinomial estimator. Ann. Mathem. Statist. 27, 642–669.MathSciNetCrossRefGoogle Scholar
  5. 5.
    Etemadi, N. (1985). Tail probabilities for sums of independent Banach space valued random variables. Sankhya 47A, 209–214.MathSciNetzbMATHGoogle Scholar
  6. 6.
    Gänssler, P. (1983). Empirical processes. Institute of Mathematical Statistics Lecture Notes Monograph Series, Vol. 3.Google Scholar
  7. 7.
    Giné, E. (1996). Lectures on some aspects of the bootstrap. Lect. Notes in Math. 1665, 37–151.MathSciNetCrossRefGoogle Scholar
  8. 8.
    Hoffmann-Jørgensen, J. (1985). The law of large numbers. Asterisque, Vol. 131.Google Scholar
  9. 9.
    Lai, T. L. (1974). Convergence rates in the strong law of large numbers for random variables taking values in Banach spaces. Bulletin of the institute of mathematics academia sinica, Vol. 2, pp. 67–85.MathSciNetzbMATHGoogle Scholar
  10. 10.
    Loéve, M. (1977). Probability Theory I, Fourth Edition, New York.Google Scholar
  11. 11.
    Montgomery-Smith, S. J. (1994). Comparison of sums of independent identically distributed random vectors. Prob. Math. Statist. 14, 281–285.MathSciNetzbMATHGoogle Scholar
  12. 12.
    Prokhorov, Yu. V. (1950). On the strong law of large numbers. Izv. Akad. Nauk. SSSR. Ser. Math. 14, 523–536.MathSciNetzbMATHGoogle Scholar
  13. 13.
    Spitzer, F. (1956). A combinatorial lemma and its applications to probability theory. Trans. Amer. Math. Soc. 82, 323–339.MathSciNetCrossRefGoogle Scholar
  14. 14.
    Talagrand, M. (1987). The Glivenko-Cantelli problem. Ann. Prob. 15, 837–870.MathSciNetCrossRefGoogle Scholar
  15. 15.
    van der Vaart, A., and Wellner, J. (1996). Weak Convergence and Empirical Processes, Springer Series in Statistics, New York.CrossRefGoogle Scholar

Copyright information

© Plenum Publishing Corporation 2000

Authors and Affiliations

  • Berthold Wittje

There are no affiliations available

Personalised recommendations