Skip to main content
SpringerLink
Log in

Complete convergence and strong laws of large numbers for weighted sums of negatively orthant dependent random variables

  • Published:
Acta Mathematica Hungarica Aims and scope Submit manuscript

Abstract

Let \({\{X_n, n\geqq 1\}}\) be a sequence of identically distributed negatively orthant dependent random variables and let \({\{a_{ni}, 1 \leqq i \leqq n, n \geqq 1\}}\) be an array of constants satisfying \({\sum_{i=1}^n |a_{ni} |^\alpha = O(n)}\) for some \({0 < \alpha < 2}\). Set \({b_n = n^{1/\alpha}({\rm log} n)^{1/\gamma}}\) and \({\gamma > \alpha}\). We give necessary and sufficient conditions for complete convergence of the form

$$\sum_{n=1}^\infty n^{-1} P \Big ( \max_{1\leqq m\leqq n} \Big| \sum_{i=1}^m a_{ni}X_i \Big|>\varepsilon b_n\Big)<\infty, \quad \forall \varepsilon>0.$$

As corollaries, strong laws of large numbers for weighted sums are obtained.

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. Asadian N., Fakoor V., Bozorgnia A.: Rosenthal’s type inequalities for negatively orthant dependent random variables. J. Iranian Stat. Soc. 5, 69–75 (2006)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Chen P., Gan S.: Limiting behavior of weighted sums of i.i.d. random variables. Stat. Probab. Lett. 77, 1589–1599 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chen P., Sung S. H.: On the strong convergence for weighted sums of negatively associated random variables. Stat. Probab. Lett. 92, 45–52 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. H. Huang and D. Wang, A note on the strong limit theorem for weighted sums of sequences of negatively dependent random variables, J. Inequal. Appl., 2012 (2012), 233.

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

    Article  MathSciNet  MATH  Google Scholar 

  9. Kuczmaszewska A.: On complete convergence for arrays of rowwise negatively associated random variables. Stat. Probab. Lett. 79, 116–124 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Kuczmaszewska, On complete convergence in Marcinkiewicz–Zygmund type SLLN for negatively associated random variables, Acta Math. Hungar., 128 (2010), 116–130.

  11. A. Kuczmaszewska and Z. A. Lagodowski, Convergence rates in the SLLN for some classes of dependent random fields, J. Math. Anal. Appl., 380 (2011), 571–584.

  12. E. L. Lehmann, Some concepts of dependence, Ann. Math. Statist., 37 (1966), 1137–1153.

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

    Article  MATH  Google Scholar 

  14. Móricz F.: Moment inequalities and the strong laws of large numbers. Z. Wahrsch. Verw. Gebiete 35, 299–314 (1976)

    Article  MATH  Google Scholar 

  15. Shen A.: On the strong convergence rate for weighted sums of arrays of rowwise negatively orthant dependent random variables. Rev. R. Acad. Cien. Serie A. Mat. 107, 257–271 (2013)

    Article  MATH  Google Scholar 

  16. Sung S. H.: Strong laws for weighted sums of i.i.d. random variables. Stat. Probab. Lett. 52, 413–419 (2001)

    Article  MATH  Google Scholar 

  17. Sung S. H.: On the strong convergence for weighted sums of random variables. Stat. Papers, 52, 447–454 (2011)

    Article  MATH  Google Scholar 

  18. X. Wang, S. Hu and W. Yang, Complete convergence for arrays of rowwise negatively orthant dependent random variables, Rev. R. Acad. Cien. Serie A. Mat., 106 (2012), 235–245.

  19. W. B. Wu, On the strong convergence of a weighted sum, Stat. Probab. Lett., 44 (1999), 19–22.

  20. Q. Zhang and D. Wang, A note on the rate of strong convergence for weighted sums of arrays of rowwise negatively orthant dependent random variables, Discrete Dyn. Nat. Soc., 2014 (2014), 368702, 7 pp.

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Jinan University, 510630, Guangzhou, P.R. China

    P. Chen

  2. Department of Applied Mathematics, Pai Chai University, 302-735, Taejon, South Korea

    S. H. Sung

Authors
  1. P. Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. H. Sung
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. H. Sung.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, P., Sung, S.H. Complete convergence and strong laws of large numbers for weighted sums of negatively orthant dependent random variables. Acta Math. Hungar. 148, 83–95 (2016). https://doi.org/10.1007/s10474-015-0559-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10474-015-0559-9

Key words and phrases

Mathematics Subject Classification

Search

Navigation