Advertisement

Acta Mathematica Sinica, English Series

, Volume 34, Issue 10, pp 1531–1548 | Cite as

Complete Moment Convergence for Arrays of Rowwise Widely Orthant Dependent Random Variables

  • Yi Wu
  • Xue Jun Wang
  • Andrew Rosalsky
Article

Abstract

In this paper, complete moment convergence for widely orthant dependent random variables is investigated under some mild conditions. For arrays of rowwise widely orthant dependent random variables, the main results extend recent results on complete convergence to complete moment convergence. These results on complete moment convergence are shown to yield new results on complete integral convergence.

Keywords

Complete convergence complete moment convergence array of widely orthant dependent random variables complete integral convergence 

MR(2010) Subject Classification

60F15 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

The authors are grateful to the referee for carefully reading the manuscript and for providing helpful comments and constructive criticism which enabled them to improve the paper.

References

  1. [1]
    Chen, Y., Wang, L., Wang, Y. B.: Uniform asymptotics for the finite-time ruin probabilities of two kinds of nonstandard bidimensional risk models. J. Math. Anal. Appl., 401, 114–129 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    Chow, Y. S.: On the rate of moment convergence of sample sums and extremes. Bull. Inst. Math. Acad. Sin., 16(3), 177–201 (1988)MathSciNetzbMATHGoogle Scholar
  3. [3]
    Hsu, P. L., Robbins, H.: Complete convergence and the law of large numbers. Proc. Natl. Acad. Sci. U.S.A., 33, 25–31 (1947)MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Hu, T. C., Rosalsky, A., Wang, K. L.: Complete convergence theorems for extended negatively dependent random variables. Sankhyā Ser. A, 77(1), 1–29 (2015) (Erratum: Sankhyā Ser. A, 78(2), 347–348 (2016))MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Hu, T. Z.: Negatively superadditive dependence of random variables with applications. Chinese J. Appl. Probab. Statist., 16, 133–144 (2000)MathSciNetzbMATHGoogle Scholar
  6. [6]
    Joag-Dev, K., Proschan, F.: Negative association of random variables with applications. Ann. Statist., 11, 286–295 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Lehmann, E. L.: Some concepts of dependence. Ann. Math. Statist., 37, 1137–1153 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Liu, L.: Precise large deviations for dependent random variables with heavy tails. Statist. Probab. Lett., 79, 1290–1298 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Qiu, D. H., Chen, P. Y.: Complete and complete moment convergence for weighted sums of widely orthant dependent random variables. Acta Math. Sin. Engl. Ser., 30, 1539–1548 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Shen, A. T.: Bernstein-type inequality for widely dependent sequence and its application to nonparametric regression models. Abstr. Appl. Anal., 2013, Article ID 862602, 9 pages (2013)Google Scholar
  11. [11]
    Shen, A. T.: On asymptotic approximation of inverse moments for a class of nonnegative random variables. Statistics, 48, 1371–1379 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Shen, A. T., Yao, M., Wang, W. J., et al.: Exponential probability inequalities for WNOD random variables and their applications. RACSAM, 110, 251–268 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    Wang, K. Y., Wang, Y. B., Gao, Q. W.: Uniform asymptotics for the finite-time ruin probability of a new dependent risk model with a constant interest rate. Methodol. Comput. Appl. Probab., 15, 109–124 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    Wang, X. J., Xu, C., Hu, T. C., et al.: On complete convergence for widely orthant dependent random variables and its applications in nonparametric regression models. TEST, 23, 607–629 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    Wang, X. J., Wu, Y., Rosalsky, A.: Complete convergence for arrays of rowwise widely orthant dependent random variables and its applications. Stochastics, 89, 1228–1252 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Wang, Y. B., Cheng, D. Y.: Basic renewal theorems for random walks with widely dependent increments. J. Math. Anal. Appl., 384, 597–606 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    Yang, W. Z., Liu, T. T., Wang, X. J., et al.: On the Bahadur representation of sample quantiles for widely orthant dependent sequences. Filomat, 28, 1333–1343 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematical SciencesAnhui UniversityHefeiP. R. China
  2. 2.Department of StatisticsUniversity of FloridaGainesvilleUSA

Personalised recommendations