Lithuanian Mathematical Journal

, Volume 54, Issue 2, pp 220–228 | Cite as

A note on the rates of convergence for weighted sums of ρ*-mixing random variables

Article

Abstract

We discuss the rates of convergence for weighted sums of ρ-mixing random variables. We solve an open problem posed by Sung [S.H. Sung, On the strong convergence for weighted sums of ρ-mixing random variables, Stat. Pap., 54:773–781, 2013]. In addition, the two obtained lemmas in this paper improve the corresponding ones of Sung in the above-mentioned paper and [S.H. Sung, On the strong convergence for weighted sums of random variables, Stat. Pap., 52:447–454, 2011].

Keywords

complete convergence weighted sums ρ-mixing random variables 

MSC

60F15 

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References

  1. 1.
    J. An and D.M. Yuan, Complete convergence of weighted sums for \( \widetilde{\rho } \)-mixing sequence of random variables, Stat. Probab. Lett., 78:1466–1472, 2008.CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    R.C. Bradley, Equivalent mixing conditions for random fields, Technical Report No. 336, Center for Stochastic Processes, Department of Statistics, University of North Carolina, Chapel Hill, 1990.Google Scholar
  3. 3.
    G.H. Cai, Marcinkiewicz strong laws for linear statistics of \( \widetilde{\rho } \)-mixing sequences of random variables, An. Acad. Bras. Ciênc., 78:615–621, 2006.CrossRefMATHGoogle Scholar
  4. 4.
    G.H. Cai, Moment inequality and complete convergence of \( \widetilde{\rho } \)-mixing sequences, J. Syst. Sci. Math. Sci., 28:251–256, 2008 (in Chinese).MATHGoogle Scholar
  5. 5.
    S.X. Gan, Almost sure convergence for \( \widetilde{\rho } \)-mixing random variable sequences, Stat. Probab. Lett., 67:289–298, 2004.CrossRefMATHGoogle Scholar
  6. 6.
    M.L. Guo and D.J. Zhu, Equivalent conditions of complete moment convergence of weighted sums for ρ -mixing sequence of random variables, Stat. Probab. Lett., 83:13–20, 2013.CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    P.L. Hsu and H. Robbins, Complete convergence and the law of large numbers, Proc. Natl. Acad. Sci. USA, 33:25–31, 1947.CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    K. Joag-Dev and F. Proschan, Negative association of random variables with applications, Ann. Stat., 11:286–295, 1983.CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    A. Kuczmaszewska, On complete convergence for arrays of rowwise dependent random variables, Stat. Probab. Lett., 77:1050–1060, 2007.CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    C.C. Moore, The degree of randomness in a stationary time series, Ann. Math. Stat., 34:1253–258, 1963.CrossRefMATHGoogle Scholar
  11. 11.
    M. Peligrad and A. Gut, Almost-sure results for a class of dependent random variables, J. Theor. Probab., 12:87–104, 1999.CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Q.M. Shao, A moment inequality and its application, Acta Math. Sin., Chin. Ser., 31:736–747, 1988 (in Chinese).MATHGoogle Scholar
  13. 13.
    S.H. Sung, On the strong convergence for weighted sums of random variables, Stat. Pap., 52:447–454, 2011.CrossRefMATHGoogle Scholar
  14. 14.
    S.H. Sung, On the strong convergence for weighted sums of ρ -mixing random variables, Stat. Pap., 54:773–781, 2013.CrossRefMATHGoogle Scholar
  15. 15.
    S. Utev and M. Peligrad, Maximal inequalities and an invariance principle for a class of weakly dependent random variables, J. Theor. Probab., 16:101–115, 2003.CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    X.J. Wang, S.H. Hu, W.Z. Yang, and Y. Shen, On complete convergence for weighted sums of φ-mixing random variables, J. Inequal. Appl., 2010, doi:10.1155/2010/372390.MathSciNetGoogle Scholar
  17. 17.
    X.J. Wang, X.Q. Li, W.Z. Yang, and S.H. Hu, On complete convergence for arrays of rowwise weakly dependent random variables, Appl. Math. Lett., 25:1916–1920, 2012.CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Q.Y. Wu and Y.Y. Jiang, Some strong limit theorems for \( \widetilde{\rho } \)-mixing sequences of random variables, Stat. Probab. Lett., 78:1017–1023, 2008.CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Y.F. Wu, C.H. Wang, and A. Volodin, Limiting behavior for arrays of rowwise ρ -mixing random variables, Lith. Math. J., 52:214–221, 2012.CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    X.C. Zhou, C.C. Tan, and J.G. Lin, On the strong laws for weighted sums of ρ -mixing random variables, J. Inequal. Appl., 2011, doi:10.1155/2011/157816.MathSciNetGoogle Scholar
  21. 21.
    M.H. Zhu, Strong laws of large numbers for arrays of rowwise ρ -mixing random variables, Discrete Dyn. Nat. Soc., 2007, Article ID 74296, 6pp., 2007.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.College of Mathematics and Computer ScienceTongling UniversityTonglingPR China
  2. 2.Department of Applied MathematicsPai Chai UniversityTaejonSouth Korea
  3. 3.Department of Mathematics and StatisticsUniversity of ReginaReginaCanada

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