Skip to main content
SpringerLink
Log in

Complete moment and integral convergence for sums of negatively associated random variables

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

For a sequence of identically distributed negatively associated random variables “X n ; n ≥ 1” with partial sums S n = Σ ni=1 X i , n ≥ 1, refinements are presented of the classical Baum-Katz and Lai complete convergence theorems. More specifically, necessary and sufficient moment conditions are provided for complete moment convergence of the form

$$ \sum\limits_{n \geqslant n_0 } {n^{r - 2 - \tfrac{1} {{pq}}} a_n E\left( {\mathop {\max }\limits_{1 \leqslant k \leqslant n} \left| {S_k } \right|^{\tfrac{1} {q}} - \varepsilon b_n^{\tfrac{1} {{pq}}} } \right)^ + < \infty } $$

to hold where r > 1, q > 0 and either n 0 = 1, 0 < p < 2, a n = 1, b n = n or n 0 = 3, p = 2, a n = (log n)−1/2q, b n = n log n. These results extend results of Chow and of Li and Spătaru from the independent and identically distributed case to the identically distributed negatively associated setting. The complete moment convergence is also shown to be equivalent to a form of complete integral convergence.

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  5. Spitzer, F.: A combinatorial lemma and its application to probability theory. Trans. Amer. Math. Soc., 82, 323–339 (1956)

    Article  MATH  MathSciNet  Google Scholar 

  6. Chow, Y. S.: On the rate of moment convergence of sample sums and extremes. Bull. Inst. Math. Acad. Sinica, 16, 177–201 (1988)

    MATH  MathSciNet  Google Scholar 

  7. Hoffmann-Jørgensen, J.: Sums of independent Banach space valued random variables. Studia Math., 52, 159–186 (1974)

    MathSciNet  Google Scholar 

  8. Li, D., Spătaru, A.: Refinement of convergence rates for tail probabilities. J. Theoret. Probab., 18, 933–947 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  9. Lai, T. L.: Limit theorems for delayed sums. Ann. Probab., 2, 432–440 (1974)

    Article  MATH  Google Scholar 

  10. Alam, K., Saxena, K. M. L.: Positive dependence in multivariate distributions. Comm. Statist. A — Theory Methods, 10, 1183–1196 (1981)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  12. Newman, C. M.: Asymptotic independence and limit theorems for positively and negatively dependent random variables. In: Tong, Y. L. (Ed.), Inequalities in Statistics and Probability: Proceedings of the Symposium on Inequalities in Statistics and Probability, October 27–30, 1982, Lincoln, Nebraska. Institute of Mathematical Statistics Lecture Notes — Monogr. Ser., 5, Institute of Mathematical Statistics, Hayward, California, 127–140, 1984

    Chapter  Google Scholar 

  13. 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 

  14. Su, C., Chi, X.: Some results on the central limit theorem for nonstationary NA sequences (in Chinese). Acta Math. Appl. Sinica, 21, 9–21 (1998)

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  16. Chi, X., Su, C.: A weak law of large numbers for negatively associated identically distributed sequences (in Chinese). Chinese J. Appl. Probab. Statist., 13, 199–203 (1997)

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  18. Liu, J., Gan, S., Chen, P.: The Hájek-Rényi inequality for NA random variables and its application. Statist. Probab. Lett., 43, 99–105 (1999), Erratum: Statist. Probab. Lett., 44, 210 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  19. Shao, Q.-M.: A comparison theorem on moment inequalities between negatively associated and independent random variables. J. Theoret. Probab., 13, 343–356 (2000)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  22. Liang, H.-Y.: Complete convergence for weighted sums of negatively associated random variables. Statist. Probab. Lett., 48, 317–325 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  23. Baek, J.-I., Kim, T.-S., Liang, H.-Y.: On the convergence of moving average processes under dependent conditions. Austral. New Zealand J. Statist., 45, 331–342 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  24. Baek, J.-I., Choi, I.-B., Niu, S.-L.: On complete convergence of weighted sums for arrays of negatively associated random variables. J. Korean Statist. Soc., 37, 73–80 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  25. Liang, H.-Y., Baek, J.-I.: Weighted sums of negatively associated random variables. Austral. New Zealand J. Statist., 48, 21–31 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  26. Wang, D. C., Su, C.: Moment complete convergence for sequences of B-valued iid random elements (in Chinese). Acta Math. Appl. Sinica, 27, 440–448 (2004)

    MATH  MathSciNet  Google Scholar 

  27. Wang, Y., Zhong, S. M., Wang, D. C.: Moment complete convergence for sums of i.i.d. random elements in a Banach space (in Chinese). Dianzi Keji Daxue Xuebao, 34, 410–412 (2005)

    MATH  MathSciNet  Google Scholar 

  28. Chen, P.: Complete moment convergence for sequences of independent random elements in Banach spaces. Stochastic Anal. Appl., 24, 999–1010 (2006)

    Article  MATH  Google Scholar 

  29. Guo, M. L., Xu, J.: Moment complete convergence for arrays of B-valued random elements (in Chinese). Appl. Math. J. Chinese Univ. Ser. A, 21, 432–438 (2006)

    MATH  MathSciNet  Google Scholar 

  30. Rosalsky, A., Thanh, L. V., Volodin, A. I.: On complete convergence in mean of normed sums of independent random elements in Banach spaces. Stochastic Anal. Appl., 24, 23–35 (2006)

    Article  MATH  Google Scholar 

  31. Ye, F., Zhu, D. J.: Moment complete convergence for B-valued random elements (in Chinese). J. Hefei Univ. Technol. Nat. Sci., 30, 525–528 (2007)

    MATH  MathSciNet  Google Scholar 

  32. Wang, D. C., Zhao, W.: Moment complete convergence for sums of a sequence of NA random variables (in Chinese). Appl. Math. J. Chinese Univ. Ser. A, 21, 445–450 (2006)

    MATH  MathSciNet  Google Scholar 

  33. Jiang, Y., Zhang, L. X.: Precise asymptotics in complete moment convergence of i.i.d. random variables (in Chinese). Acta Math. Sci. Ser. A Chin. Ed., 26, 917–925 (2006)

    MATH  MathSciNet  Google Scholar 

  34. Liu, W., Lin, Z.: Precise asymptotics for a new kind of complete moment convergence. Statist. Probab. Lett., 76, 1787–1799 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  35. Ye, J., Zhang, L.-X., Pang, T.-X.: Precise rates in the law of logarithm for the moment convergence of i.i.d. random variables. J. Math. Anal. Appl., 327, 695–714 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  36. Fu, K.-A., Zhang, L.-X.: On the moment convergence rates of LIL in Hilbert space. Math. Comput. Modelling, 47, 153–167 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  37. Jiang, Y., Zhang, L. X.: Precise asymptotics in the complete moment convergence of NA sequences (in Chinese). J. Zhejiang Univ. Sci. Ed., 32, 490–493 (2005)

    MATH  MathSciNet  Google Scholar 

  38. Spătaru, A.: Strengthening the Hsu-Robbins-Erdös theorem. Rev. Roumaine Math. Pures Appl., 35, 463–465 (1990)

    MATH  MathSciNet  Google Scholar 

  39. Spătaru, A.: Refinement of convergence rates for tail probabilities. An. Univ. Craiova Ser. Mat. Inform., 32, 221–226 (2005)

    MATH  MathSciNet  Google Scholar 

  40. Spătaru, A.: Strengthening classical results on convergence rates in strong limit theorems. Probab. Theory Rel. Fields, 136, 1–18 (2006)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Tongji University, Shanghai, 200092, P. R. China

    Han Ying Liang

  2. Department of Mathematical Sciences, Lakehead University, Thunder Bay, Ontario, P7B 5E1, Canada

    De Li Li

  3. Department of Statistics, University of Florida, Gainesville, Florida, 32611, USA

    Andrew Rosalsky

Authors
  1. Han Ying Liang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. De Li Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Andrew Rosalsky
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Han Ying Liang.

Additional information

The first author is supported by National Natural Science Foundation of China (Grant No. 10871146) and the second author is supported by Natural Sciences and Engineering Research Council of Canada

Rights and permissions

Reprints and permissions

About this article

Cite this article

Liang, H.Y., Li, D.L. & Rosalsky, A. Complete moment and integral convergence for sums of negatively associated random variables. Acta. Math. Sin.-English Ser. 26, 419–432 (2010). https://doi.org/10.1007/s10114-010-8177-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-010-8177-5

Keywords

MR(2000) Subject Classification

Search

Navigation