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Complete moment convergence for weighted sums of extended negatively dependent random variables

  • School of Mathematical Sciences,Anhui University,Hefei 230601,China

Cite this:
https://doi.org/10.3969/j.issn.0253-2778.2020.02.011
  • Received Date: 31 July 2019
  • Accepted Date: 20 November 2019
  • Rev Recd Date: 20 November 2019
  • Publish Date: 28 February 2020
    Abstract

  • Abstract

    The complete convergence and complete moment convergence for weighted sums of extended negatively dependent (END) random variables was studied by using the Rosenthal type moment inequality and truncation method of random variables.The obtained results extend some corresponding ones for independent random variables and negatively associated (NA) random variables in the existing literature.

    Abstract

    The complete convergence and complete moment convergence for weighted sums of extended negatively dependent (END) random variables was studied by using the Rosenthal type moment inequality and truncation method of random variables.The obtained results extend some corresponding ones for independent random variables and negatively associated (NA) random variables in the existing literature.
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    • References(22)
  • [1]
    ALAM K,SAXENA K M L.Positive dependence in multivariate distributions[J].Communications in Statistics: Theory and Methods,1981,10: 1183-1196.
    [2]
    JOAG-DEV K,PROSCHAN F.Negative association of random variables with applications[J].The Annals of Statistics,1983,11: 286-295.
    [3]
    LIU L.Precise large deviations for dependent random variables with heavy tails[J].Statistics and Probability Letters,2009,79: 1290-1298.
    [4]
    HU T Z.Negatively superadditive dependence of random variables with applications[J].Chinese Journal of Applied Probability and Statistics,2000,16: 133-144.
    [5]
    LIU L.Necessary and sufficient conditions for moderate deviations of dependent random variables with heavy tails[J].Science China Mathematics,2010,53(6): 1421-1434.
    [6]
    SHEN A T.Probability inequalities for END sequence and their applications[J].Journal of Inequalities and Applications,2011,2011: 98.
    [7]
    WU Y F,GUAN M.Convergence properties of the partial sums for sequences of END random variables[J].Journal of the Korean Mathematical Society,2012,49(6): 1097-1110.
    [8]
    QIU D H,CHEN P Y,ANTONINI R G,et al.On the complete convergence for arrays of rowwise extended negatively dependent random variables[J].Journal of the Korean Mathematical Society,2013,50(2): 379-392.
    [9]
    WANG X J,HU S H,HU T C.Complete convergence for weighted sums and arrays of rowwise END sequences[J].Communications in Statistics: Theory and Methods,2013,42: 2391-2401.
    [10]
    WANG S J,WANG X J.Precise large deviations for random sums of END real-valued random variables with consistent variation[J].Journal of Mathematical Analysis and Applications,2013,402: 660-667.
    [11]
    WANG X J,ZHENG L L,XU C,et al.Complete consistency for the estimator of nonparametric regression models based on extended negatively dependent errors[J].Statistics: A Journal of Theoretical and Applied Statistics,2015,49(2): 396-407.
    [12]
    SHEN A T.Complete convergence for weighted sums of END random variables and its application to nonparametric regression models[J].Journal of Nonparametric Statistics,2016,28(4): 702-715.
    [13]
    SHEN A T,VOLODIN A.Weak and strong laws of large numbers for arrays of rowwise END random variables and their applications[J].Metrika,2017,80: 605-625.
    [14]
    SHEN A T,YAO M,XIAO B Q.Equivalent conditions of complete convergence and complete moment convergence for END random variables[J].Chinese Annals of Mathematics,Series B,2018,39(1): 83-96.
    [15]
    WU Y F,CABREA M O,VOLODIN A.Complete convergence and complete moment convergence for arrays of rowwise END random variables[J].Glasnik Matematicki,2014,49(2):
    449-468.
    [16]
    YANG W Z,XU H Y,CHEN L,et al.Complete consistency of estimators for regression models based on extended negatively dependent errors[J].Statistical Papers,2018,59(2): 449-465.
    [17]
    HSU P L,ROBBINS H.Complete convergence and law of large numbers[J].Proceedings of the National Academy of Sciences of the United States of America,1947,33(2): 25-31.
    [18]
    LI D L,RAO M B,JIANG T F,et al.Complete convergence and almost sure convergence of weighted sums of random variables[J].Journal of Theoretical Probability,1995,8: 49-76.
    [19]
    LIANG H Y.Complete convergence for weighted sums of negatively associated random variables[J].Statistics and Probability Letters,2000,48: 317-325.
    [20]
    CHEN P Y,SUNG S H.On complete convergence and complete moment convergence for weighted sums of ρ*-mixing random variables[J].Journal of Inequalities and Applications,2018,2018: 121.
    [21]
    CHOW Y S.On the rate of moment convergence of sample sums and extremes[J].Bulletin of the Institute of Mathematics Academia Sinica,1988,16: 177-201.
    [22]
    吴群英.混合序列的概率极限理论[M].北京：科学出版社，2006.)
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    [1]
    ALAM K,SAXENA K M L.Positive dependence in multivariate distributions[J].Communications in Statistics: Theory and Methods,1981,10: 1183-1196.
    [2]
    JOAG-DEV K,PROSCHAN F.Negative association of random variables with applications[J].The Annals of Statistics,1983,11: 286-295.
    [3]
    LIU L.Precise large deviations for dependent random variables with heavy tails[J].Statistics and Probability Letters,2009,79: 1290-1298.
    [4]
    HU T Z.Negatively superadditive dependence of random variables with applications[J].Chinese Journal of Applied Probability and Statistics,2000,16: 133-144.
    [5]
    LIU L.Necessary and sufficient conditions for moderate deviations of dependent random variables with heavy tails[J].Science China Mathematics,2010,53(6): 1421-1434.
    [6]
    SHEN A T.Probability inequalities for END sequence and their applications[J].Journal of Inequalities and Applications,2011,2011: 98.
    [7]
    WU Y F,GUAN M.Convergence properties of the partial sums for sequences of END random variables[J].Journal of the Korean Mathematical Society,2012,49(6): 1097-1110.
    [8]
    QIU D H,CHEN P Y,ANTONINI R G,et al.On the complete convergence for arrays of rowwise extended negatively dependent random variables[J].Journal of the Korean Mathematical Society,2013,50(2): 379-392.
    [9]
    WANG X J,HU S H,HU T C.Complete convergence for weighted sums and arrays of rowwise END sequences[J].Communications in Statistics: Theory and Methods,2013,42: 2391-2401.
    [10]
    WANG S J,WANG X J.Precise large deviations for random sums of END real-valued random variables with consistent variation[J].Journal of Mathematical Analysis and Applications,2013,402: 660-667.
    [11]
    WANG X J,ZHENG L L,XU C,et al.Complete consistency for the estimator of nonparametric regression models based on extended negatively dependent errors[J].Statistics: A Journal of Theoretical and Applied Statistics,2015,49(2): 396-407.
    [12]
    SHEN A T.Complete convergence for weighted sums of END random variables and its application to nonparametric regression models[J].Journal of Nonparametric Statistics,2016,28(4): 702-715.
    [13]
    SHEN A T,VOLODIN A.Weak and strong laws of large numbers for arrays of rowwise END random variables and their applications[J].Metrika,2017,80: 605-625.
    [14]
    SHEN A T,YAO M,XIAO B Q.Equivalent conditions of complete convergence and complete moment convergence for END random variables[J].Chinese Annals of Mathematics,Series B,2018,39(1): 83-96.
    [15]
    WU Y F,CABREA M O,VOLODIN A.Complete convergence and complete moment convergence for arrays of rowwise END random variables[J].Glasnik Matematicki,2014,49(2):
    449-468.
    [16]
    YANG W Z,XU H Y,CHEN L,et al.Complete consistency of estimators for regression models based on extended negatively dependent errors[J].Statistical Papers,2018,59(2): 449-465.
    [17]
    HSU P L,ROBBINS H.Complete convergence and law of large numbers[J].Proceedings of the National Academy of Sciences of the United States of America,1947,33(2): 25-31.
    [18]
    LI D L,RAO M B,JIANG T F,et al.Complete convergence and almost sure convergence of weighted sums of random variables[J].Journal of Theoretical Probability,1995,8: 49-76.
    [19]
    LIANG H Y.Complete convergence for weighted sums of negatively associated random variables[J].Statistics and Probability Letters,2000,48: 317-325.
    [20]
    CHEN P Y,SUNG S H.On complete convergence and complete moment convergence for weighted sums of ρ*-mixing random variables[J].Journal of Inequalities and Applications,2018,2018: 121.
    [21]
    CHOW Y S.On the rate of moment convergence of sample sums and extremes[J].Bulletin of the Institute of Mathematics Academia Sinica,1988,16: 177-201.
    [22]
    吴群英.混合序列的概率极限理论[M].北京：科学出版社，2006.)
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