Skip to main content
SpringerLink
Log in

Strong Limit Theorems for Weighted Sums of Negatively Associated Random Variables

  • Published:
Journal of Theoretical Probability Aims and scope Submit manuscript

Abstract

In this paper, we establish strong laws for weighted sums of identically distributed negatively associated random variables. Marcinkiewicz-Zygmund’s strong law of large numbers is extended to weighted sums of negatively associated random variables. Furthermore, we investigate various limit properties of Cesàro’s and Riesz’s sums of negatively associated random variables. Some of the results in the i.i.d. setting, such as those in Jajte (Ann. Probab. 31(1), 409–412, 2003), Bai and Cheng (Stat. Probab. Lett. 46, 105–112, 2000), Li et al. (J. Theor. Probab. 8, 49–76, 1995) and Gut (Probab. Theory Relat. Fields 97, 169–178, 1993) are also improved and extended to the negatively associated setting.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alam, K., Saxena, K.M.L.: Positive dependence in multivariate distributions. Commun. Stat. Theory Methods A10, 1183–1196 (1981)

    MathSciNet  Google Scholar 

  2. Baek, J.I., Kim, T.S., Liang, H.Y.: On the convergence of moving average processes under dependent conditions. Aust. N. Z. J. Stat. 45, 331–342 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bai, Z.D., Cheng, P.E.: Marcinkiewicz strong laws for linear statistics. Stat. Probab. Lett. 46, 105–112 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bai, Z.D., Cheng, P.E., Zhang, C.H.: An extension of the Hardy-Littlewood strong law. Stat. Sinica 7, 923–928 (1997)

    MATH  MathSciNet  Google Scholar 

  5. Baum, L.E., Katz, M.: Convergence rates in the law of large numbers. Trans. Am. Math. Soc. 120, 108–123 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bingham, N.H., Nili Sant, H.R.: Summability methods and negatively associated random variables. J. Appl. Probab. 41A, 231–238 (2004)

    Article  MATH  Google Scholar 

  7. Bingham, N.H., Tenenbaum, M.: Riesz and Valiron means and fractional moments. Math. Proc. Camb. Philos. Soc. 99, 143–149 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  8. Chandra, T.K., Ghosal, S.: Extensions of the strong law of large numbers of Marcinkiewicz and Zygmund for dependent variables. Acta Math. Hung. 71, 327–336 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  9. Chow, Y.S.: Some convergence theorems for independent random variables. Ann. Math. Stat. 37, 1482–1492 (1966)

    Article  MATH  Google Scholar 

  10. Chow, Y.S.: Delayed sums and Borel summability of independent identically distributed random variables. Bull. Inst. Math. Acad. Sinica 1, 207–220 (1973)

    MATH  MathSciNet  Google Scholar 

  11. Chow, Y.S., Lai, T.L.: Limiting behavior of weighted sums independent random variables. Ann. Probab. 1, 810–824 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  12. Cuzick, J.: A strong law for weighted sums of i.i.d. random variables. J. Theor. Probab. 8, 625–641 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  13. Déniel, Y., Derriennic, Y.: Sur la convergence presque sure, au sens de Cesàro d’ordre α, 0<α<1, de variables aléatoires et indépendantes et identiquement distribuées. Probab. Theory Relat. Fields 79, 629–636 (1988)

    Article  MATH  Google Scholar 

  14. Erdös, P.: On a theorem of Hsu and Robbins. Ann. Math. Stat. 20, 286–291 (1949)

    Article  MATH  Google Scholar 

  15. Erdös, P.: Remark on my paper “On a theorem of Hsu and Robbins”. Ann. Math. Stat. 21, 138 (1950)

    Article  MATH  Google Scholar 

  16. Gut, A.: Complete convergence and Cesàro summation for i.i.d. random variables. Probab. Theory Relat. Fields 97, 169–178 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  17. Heinkel, B.: An infinite-dimensional law of large numbers in Cesàro’s sense. J. Theor. Probab. 3, 533–546 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  18. Hsu, P.L., Robbins, H.: Complete convergence and the law of large numbers. Proc. Natl. Acad. Sci. USA 33, 25–31 (1947)

    Article  MATH  MathSciNet  Google Scholar 

  19. Jajte, R.: On the strong law of large numbers. Ann. Probab. 31(1), 409–412 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  20. Joag-Dev, K., Proschan, F.: Negative association of random variables with applications. Ann. Stat. 11, 286–295 (1983)

    Article  MathSciNet  Google Scholar 

  21. Lai, T.L.: Summability methods for independent identically distributed random variables. Proc. Am. Math. Soc. 45, 253–261 (1974)

    Article  MATH  Google Scholar 

  22. Lanzinger, H., Stadtmüler, U.: Weighted sums for i.i.d. random variables with relatively thin tails. Bernoulli 6, 45–61 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  23. Lanzinger, H., Stadtmüller, U.: Baum-Katz laws for certain weighted sums of independent and identically distributed random variables. Bernoulli 9, 985–1002 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  24. Lehmann, E.L.: Some concepts of dependence. Ann. Math. Stat. 37, 1137–1153 (1966)

    Article  MATH  MathSciNet  Google Scholar 

  25. Li, D.L., Rao, M.B., Jiang, T.F., et al.: Complete convergence and almost sure convergence of weighted sums of random variables. J. Theor. Probab. 8, 49–76 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  26. Liang, H.Y., Baek, J.I.: Weighted sums of negatively associated random variables. Aust. N. Z. J. Stat. 48(1), 21–31 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  27. Liang, H.Y., Su, C.: Complete convergence for weighted sums of NA sequences. Stat. Probab. Lett. 45, 85–95 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  28. Lorentz, G.G.: Borel and Banach properties of methods of summation. Duke Math. J. 22, 129–141 (1955)

    Article  MATH  MathSciNet  Google Scholar 

  29. Matula, P.: A note on the almost sure convergence of sums of negatively dependent random variables. Stat. Probab. Lett. 15, 209–213 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  30. Petrov, V.V.: Limit Theorems of Probability Theory. Oxford University Press, New York (1995)

    MATH  Google Scholar 

  31. Pruss, A.R.: A two-sided estimate in the Hsu-Robbins-Erdös law of large numbers. Stoch. Process. Appl. 70, 173–180 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  32. Pruss, A.R.: A general Hsu-Robbins-Erdös type estimate of tail probabilities of sums of independent identically distributed random variables. Period. Math. Hung. 46, 181–201 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  33. Roussas, G.G.: Asymptotic normality of random fields of positively or negatively associated processes. J. Multivariate Anal. 50, 152–173 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  34. Shao, Q.M.: A comparison theorem on maximum inequalities between negatively associated and independent random variables. J. Theor. Probab. 13, 343–356 (2000)

    Article  MATH  Google Scholar 

  35. Shao, Q.M., Su, C.: The law of the iterated logarithm for negatively associated random variables. Stoch. Process. Appl. 83, 139–148 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  36. Spitzer, F.: A combinatorial lemma and its applications to probability theory. Trans. Am. Math. Soc. 82, 323–339 (1956)

    Article  MATH  MathSciNet  Google Scholar 

  37. Su, C., Wang, Y.B.: Strong convergence for IDNA sequences. Chin. J. Appl. Probab. Stat. 14(2), 131–140 (1998) (in Chinese)

    MATH  MathSciNet  Google Scholar 

  38. Su, C., Zhao, L.C., Wang, Y.B.: Moment inequalities and week convergence for negatively associated sequences. Sci. China Ser. A 40, 172–182 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  39. Thrum, R.: A remark on almost sure convergence of weighted sums. Probab. Theory Relat. Fields 75, 425–430 (1987)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Hong Kong Univ. of Science and Technology, Clear Water Bay, Kowloon, Hong Kong

    Bing-Yi Jing

  2. Department of Mathematics, Tongji University, Shanghai, 200092, People’s Republic of China

    Han-Ying Liang

Authors
  1. Bing-Yi Jing
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Han-Ying Liang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Han-Ying Liang.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jing, BY., Liang, HY. Strong Limit Theorems for Weighted Sums of Negatively Associated Random Variables. J Theor Probab 21, 890–909 (2008). https://doi.org/10.1007/s10959-007-0128-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10959-007-0128-4

Keywords

Mathematics Subject Classification (2000)

Search

Navigation