Journal of Theoretical Probability

, Volume 12, Issue 3, pp 811–819 | Cite as

Precise Asymptotics in Spitzer's Law of Large Numbers

  • Aurel Spătaru
Article

Abstract

Let X, X1, X2,... be a sequence of i.i.d. random variables such that EX=0, assume the distribution of X is attracted to a stable distribution with exponent α<1, and set Sn=X1+ ··· +Xn. We prove that
$$\sum\limits_{n{\text{ }} \geqslant {\text{ }}1} {{\text{ }}\frac{{\text{1}}}{{\text{n}}}{\text{ }}} P(|S_n | \geqslant \varepsilon n){\text{ }} \sim {\text{ }}\frac{\alpha }{{\alpha - 1}}{\text{ }}( - \log \varepsilon {\text{) as }}\varepsilon \searrow {\text{0}}$$
Tail probabilities of sums of i.i.d. random variables stable distributions Spitzer's theorem Fuk–Nagaev type inequality 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

REFERENCES

  1. 1.
    Chen, R. (1978). A remark on the tail probability of a distribution. J. Multivariate Anal. 8, 328–333.Google Scholar
  2. 2.
    Chow, Y. S., and Teicher, H. (1978). Probability Theory, Springer, New York.Google Scholar
  3. 3.
    Cramér, H. (1946). Mathematical Methods of Statistics, Princeton University Press, Princeton.Google Scholar
  4. 4.
    Erdős, P. (1949). On a theorem of Hsu and Robbins. Ann. Math. Statist. 20, 286–291.Google Scholar
  5. 5.
    Erdős, P. (1950). Remark on my paper “On a theorem of Hsu and Robbins.” Ann. Math. Statist. 21, 138.Google Scholar
  6. 6.
    Fuk, D. H., and Nagaev, S. V. (1971). Probability inequalities for sums of independent random variables. Th. Prob. Appl. 16, 643–660.Google Scholar
  7. 7.
    Heyde, C. C. (1975). A supplement to the strong law of large numbers. J. Appl. Prob. 12, 173–175.Google Scholar
  8. 8.
    Hsu, P. L., and Robbins, H. (1947). Complete convergence and the law of large numbers. Proc. Nat. Acad. Sci. U.S.A. 33, 25–31.Google Scholar
  9. 9.
    Ibragimov, I. A., and Linnik, Yu. V. (1971). Independent and Stationary Sequences of Random Variables, Wolters-Noordhoff, Groningen.Google Scholar
  10. 10.
    Katz, M. (1963). The probability in the tail of a distribution. Ann. Math. Statist. 34, 312–328.Google Scholar
  11. 11.
    Loéve, M. (1977). Probability Theory, Vol. 1. Fourth Edition, Springer, New York.Google Scholar
  12. 12.
    Pruss, A. R. (1997). A two-sided estimate in the Hsu-Robbins-Erdős law of large numbers. Stoch. Proc. Appl. 70, 173–180.Google Scholar
  13. 13.
    Pruss, A. R. (1977). Comparisons between tail probabilities of sums of independent symmetric random variables. Ann. Inst. H. Poincaré Prob. Statist. 33, 651–671.Google Scholar
  14. 14.
    Slivka, J., and Severo, N. C. (1970). On the strong law of large numbers. Proc. Amer. Math. Soc. 24, 729–734.Google Scholar
  15. 15.
    SpĂtaru, A. (1990). Strengthening the Hsu-Robbins-Erdős theorem. Rev. Roumaine Math. Pures Appl. 35, 463–465.Google Scholar
  16. 16.
    Spitzer, F. (1956). A combinatorial lemma and its applications to probability theory. Trans. Amer. Math. Soc. 82, 323–339.Google Scholar
  17. 17.
    Vuillieumier, M. (1967). Sur le comportement asymptotique des transformations linéaires des suites. Math. Zeitsch. 98, 126–139.Google Scholar
  18. 18.
    Wu, C. F. (1973). A note on the convergence rate of the strong law of large numbers. Bull. Inst. Math. Acad. Sinica 1, 121–124.Google Scholar

Copyright information

© Plenum Publishing Corporation 1999

Authors and Affiliations

  • Aurel Spătaru
    • 1
  1. 1.Centre of Mathematical StatisticsRomanian AcademyBucharest 5Romania

Personalised recommendations