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

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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].

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References

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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. 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.

    Article  MATH  Google Scholar 

  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).

    MATH  Google Scholar 

  5. S.X. Gan, Almost sure convergence for \( \widetilde{\rho } \)-mixing random variable sequences, Stat. Probab. Lett., 67:289–298, 2004.

    Article  MATH  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  9. A. Kuczmaszewska, On complete convergence for arrays of rowwise dependent random variables, Stat. Probab. Lett., 77:1050–1060, 2007.

    Article  MATH  MathSciNet  Google Scholar 

  10. C.C. Moore, The degree of randomness in a stationary time series, Ann. Math. Stat., 34:1253–258, 1963.

    Article  MATH  Google Scholar 

  11. M. Peligrad and A. Gut, Almost-sure results for a class of dependent random variables, J. Theor. Probab., 12:87–104, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  12. Q.M. Shao, A moment inequality and its application, Acta Math. Sin., Chin. Ser., 31:736–747, 1988 (in Chinese).

    MATH  Google Scholar 

  13. S.H. Sung, On the strong convergence for weighted sums of random variables, Stat. Pap., 52:447–454, 2011.

    Article  MATH  Google Scholar 

  14. S.H. Sung, On the strong convergence for weighted sums of ρ ∗-mixing random variables, Stat. Pap., 54:773–781, 2013.

    Article  MATH  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    MathSciNet  Google Scholar 

  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.

    Article  Google Scholar 

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Authors and Affiliations

  1. College of Mathematics and Computer Science, Tongling University, Tongling, 244000, PR China

    Yongfeng Wu

  2. Department of Applied Mathematics, Pai Chai University, Taejon, 302735, South Korea

    Soo Hak Sung

  3. Department of Mathematics and Statistics, University of Regina, Regina, Saskatchewan, S4S 0A2, Canada

    Andrei Volodin

Authors
  1. Yongfeng Wu
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  2. Soo Hak Sung
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  3. Andrei Volodin
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Corresponding author

Correspondence to Yongfeng Wu.

Additional information

The research of author was partially supported by the Humanities and Social Sciences Foundation for the Youth Scholars of Ministry of Education of China (12YJCZH217), the Key NSF of Anhui Educational Committee (KJ2014A255), and the Natural Science Foundation of Anhui Province (1308085MA03).

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Wu, Y., Sung, S.H. & Volodin, A. A note on the rates of convergence for weighted sums of ρ *-mixing random variables. Lith Math J 54, 220–228 (2014). https://doi.org/10.1007/s10986-014-9239-7

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  • DOI: https://doi.org/10.1007/s10986-014-9239-7

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