Skip to main content
SpringerLink
Log in

Complete and complete moment convergence for weighted sums of widely orthant dependent random variables

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

Abstract

In this paper, we establish a complete convergence result and a complete moment convergence result for weighted sums of widely orthant dependent random variables under mild conditions. As corollaries, the corresponding results for weighted sums of extended negatively orthant dependent random variables are also obtained, which generalize and improve the related known works in the literature.

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. Block, H. W., Savits, T. H., Shaked, M.: Some concept of negative dependence. Ann. Probab., 10(3), 765–772 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  2. Chen, P. Y.: Complete moment convergence for sequences of independent random elements in Banach spaces. Stoch. Anal. Appl., 24(5), 999–1010 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  3. Chen, P. Y., Hu, T.-C., Volodin, A.: Limiting behavior of moving average processes under negative association. Teor.ĬImovīr. Mat. Stat., 7, 154–166 (2007)

    Google Scholar 

  4. Chen, Y., Wang, Y. B., Wang, K. Y.: Asymptotic results for ruin probability of a two-dimensional renewal risk model. Stoch. Anal. Appl., 31(1), 80–91 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  5. Chow, Y. S.: On the rate of moment convergence of sample sums and extremes. Bull. Inst. Math. Acad. Sin., 16(3), 177–201 (1988)

    MATH  Google Scholar 

  6. Ebrahimi, N., Ghosh, M.: Multivariate negative dependence. Comm. Statist. Theory Methods, 10(4), 307–337 (1981)

    Article  MathSciNet  Google Scholar 

  7. He, W., Cheng, D. Y., Wang, Y. B.: Asymptotic lower bounds of precise large deviations with nonnegative and dependent random variables. Statist. Probab. Lett., 81, 331–338 (2012)

    MathSciNet  Google Scholar 

  8. Hsu, P. L., Robbins, H.: Complete convergence and the law of large numbers. Proc. Natl. Acad. Sci. USA, 33(2), 35–31 (1947)

    Article  MathSciNet  Google Scholar 

  9. Jing, B. Y., Liang, H. Y.: Strong limit theorems for weighted sums of negatively associated random variables. J. Theoret. Probab., 21(4), 890–909 (2008)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  11. Li, Y. X., Zhang, L. X.: Complete moment convergence of moving-average processes under dependence assumptions. Statist. Probab. Lett., 70(3), 191–197 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  12. Liang, H. Y., Li, D. L., Rosalsky, A.: Complete moment and integral convergence for sums of negatively associated random variables. Acta Math. Sin., Engl. Series, 26(3), 419–432 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  13. Liu, L.: Precise large deviations for dependent random variables with heavy tails. Statist. Probab. Lett., 79(9), 1290–1298 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  14. Liu, L.: Necessary and sufficient conditions for moderate deviations of dependent random variables with heavy tails. Sci. China Math., 53(6), 1421–1434 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  15. Liu, X. J., Gao, Q. W., Wang, Y. B.: A note on a dependent risk model with constant interest rate. Statist. Probab. Lett., 82, 707–712 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  16. Qiu, D. H., Urmeneta, H., Volodin, A.: Complete moment convergence for weighted sums of sequences of independent random elements in Banach spaces. Collect. Math., 65, 155–167 (2014)

    Article  MathSciNet  Google Scholar 

  17. Stout, W. F.: Almost Sure Convergence, Academic Press, New York, 1974

    MATH  Google Scholar 

  18. Sung, S. H.: Complete convergence for weighted sums of arrays of rowwise independent B-valued random variables. Stoch. Anal. Appl., 15(2), 255–267 (1997)

    Article  MATH  Google Scholar 

  19. Sung, S. H.: Complete convergence for weighted sums of negatively dependent random variables. Statist. Papers, 53(1), 73–82 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  20. Wang, D. C., Su, C.: Moment complete convergence for sequences of B-valued iid random elements (in Chinese). Acta Math. Appl. Sin., 27(3), 440–448 (2004)

    MATH  MathSciNet  Google Scholar 

  21. Wang, D. C., Zhao, W.: Moment complete convergence for sums of a sequence of NA random variables. Appl. Math. J. Chinese Univ. Ser. A, 21(4), 445–450 (2006)

    MATH  MathSciNet  Google Scholar 

  22. Wang, K. Y., Wang, Y. B., Gao, Q. W.: Uniform asymptotic for the finite-time ruin probability of dependent risk model with a constant interest rate. Methodol. Comput. Appl. Probab., 15(1), 109–124(2013)

  23. Wang, Y. B., Cheng, D. Y.: Basic renewal theorems for random walks with widely dependent increments. J. Math. Anal. Appl., 384, 597–606 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  24. Wang, Y. B., Su, C.: Strong limit theorems for weighted sums of NA sequences with different distributions and their application in linear models (in Chinese). Acta Math. Appl. Sin., 21(4), 571–578 (1998)

    MATH  MathSciNet  Google Scholar 

  25. Wu, Q. Y.: A strong limit theorem for weighted sums of sequences of negatively dependent random variables. J. Inequal. Appl., Article ID 383805, 8 pp., doi: 10.1155/2010/383805

  26. Yang, Y., Wang, Y. B.: Tail behavior of the product of two dependent random variables with applications to risk theory. Extremes, 16, 55–74 (2013)

    Article  MathSciNet  Google Scholar 

  27. Zarei, H., Jabbari, H.: Complete convergence of weighted sums under negatively dependence. Statist. Papers, 52(2), 413–418 (2011)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Guangdong University of Finance & Economics, Guangzhou, 510320, P. R. China

    De Hua Qiu

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

    Ping Yan Chen

Authors
  1. De Hua Qiu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ping Yan Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to De Hua Qiu.

Additional information

Supported by National Natural Science Foundation of China (Grant No. 11271161)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Qiu, D.H., Chen, P.Y. Complete and complete moment convergence for weighted sums of widely orthant dependent random variables. Acta. Math. Sin.-English Ser. 30, 1539–1548 (2014). https://doi.org/10.1007/s10114-014-3483-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-014-3483-y

Keywords

MR(2010) Subject Classification

Search

Navigation