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CN 34-1054/N

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Strong limit theorems for negatively associated random variables with general moment conditions

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

Cite this:
https://doi.org/10.3969/j.issn.0253-2778.2015.06.005
  • Received Date: 04 June 2014
  • Accepted Date: 22 September 2014
  • Rev Recd Date: 22 September 2014
  • Publish Date: 30 June 2015
    Abstract

  • Abstract

    Let {X,Xn,n≥1} be a sequence of negatively associated random variables with identical distribution, {an,n≥1} be a sequence of positive constants with an/n↑. The strong law of large numbers and complete convergence for {X,Xn,n≥1} were obtained. These results are equivalent to the general moment condition ∑∞[]n=1P(|X|>an)<∞. On the other hand, the results extend the corresponding ones for pairwise independent random variables with identical distribution.

    Abstract

    Let {X,Xn,n≥1} be a sequence of negatively associated random variables with identical distribution, {an,n≥1} be a sequence of positive constants with an/n↑. The strong law of large numbers and complete convergence for {X,Xn,n≥1} were obtained. These results are equivalent to the general moment condition ∑∞[]n=1P(|X|>an)<∞. On the other hand, the results extend the corresponding ones for pairwise independent random variables with identical distribution.
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    • References(19)
  • [1]
    Baum L E, Katz M. Convergence rates in the law of large numbers[J]. Transactions of the American Mathematical Society, 1965, 120: 108-123.
    [2]
    Alam K, Saxena K M L. Positive dependence in multivariate distributions[J]. Communications in Statistics: Theory and Methods, 1981, 12: 1 183-1 196.
    [3]
    Joag-Dev K, Proschan F. Negative association of random variables with applications[J]. The Annals of Statistics, 1983, 11: 286-295.
    [4]
    Cai G H. Strong laws of weighted sums of NA random variables[J]. Metrika, 2008, 68: 323-331.
    [5]
    Ling N X. The Bahadur representation for sample quantiles under negatively associated sequence[J]. Statistics and Probability Letters, 2008, 78: 2 660-2 663.
    [6]
    Shao Q M. A comparison theorem on moment inequalities between negatively associated and independent random variables[J]. Journal Theoretical Probability, 2000, 13: 343-356.
    [7]
    Liang H Y, Zhang J J. Strong convergence for weighted sums of negatively associated arrays[J]. Chinese Annals of Mathematics, Series B, 2010, 31(2): 273-288.
    [8]
    Wu Q Y, Jiang Y Y. Chovers law of the iterated logarithm for negatively associated sequences[J]. Journal of Systems Science and Complexity, 2010, 23: 293-302.
    [9]
    Wu Q Y, Jiang Y Y. A law of the iterated logarithm of partial sums for negatively associated random variables[J]. Journal of the Korean Statistical Society, 2010, 39: 199-206.
    [10]
    Wang X J, Hu S H, Yang W Z. Some convergence results for arbitrary sequences under moment condition[J]. Chinese Quarterly Journal of Mathematics, 2011, 26 (4): 585-589.
    [11]
    Wang Z Z. On strong law of large numbers for random sequence[J]. Chinese Quarterly Journal of Mathematics, 2010, 25 (4): 475-480.
    [12]
    Shen A T. On strong convergence for weighted sums of a class of random variables[J]. Abstract and Applied Analysis, 2013, 2013: Article ID 216236.
    [13]
    Shen A T. Bernstein-type inequality for widely dependent sequence and its application to nonparametric regression models[J]. Abstract and Applied Analysis, 2013, 2013: Article ID 862602.
    [14]
    Shen A T, Wu R C. Some probability inequalities for a class of random variables and their applications[J]. Journal of Inequalities and Applications, 2013, 2013: 57.
    [15]
    Yang S. Uniformly asymptotic normality of the regression weighted estimator for negatively associated samples[J]. Statistics and Probability Letters, 2003, 2: 101-110.
    [16]
    Matula P. A note on the almost sure convergence of sums of negatively dependent random variables[J]. Statistics and Probability Letters, 1992, 15: 209-213.
    [17]
    吴群英. 混合序列的概率极限理论[M]. 北京: 科学出版社, 2006.
    [18]
    Sung S H. On the strong law of large numbers for pairwise i.i.d. random variables with general moment conditions[J]. Statistics and Probability Letters, 2013, 83: 1 963-1 968.
    [19]
    Wang X J, Li X Q, Hu S H, et al. Strong limit theorems for weighted sums of negatively associated random variables[J]. Stochastic Analysis and Applications, 2011, 29 (1): 1-14.
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    [1]
    Baum L E, Katz M. Convergence rates in the law of large numbers[J]. Transactions of the American Mathematical Society, 1965, 120: 108-123.
    [2]
    Alam K, Saxena K M L. Positive dependence in multivariate distributions[J]. Communications in Statistics: Theory and Methods, 1981, 12: 1 183-1 196.
    [3]
    Joag-Dev K, Proschan F. Negative association of random variables with applications[J]. The Annals of Statistics, 1983, 11: 286-295.
    [4]
    Cai G H. Strong laws of weighted sums of NA random variables[J]. Metrika, 2008, 68: 323-331.
    [5]
    Ling N X. The Bahadur representation for sample quantiles under negatively associated sequence[J]. Statistics and Probability Letters, 2008, 78: 2 660-2 663.
    [6]
    Shao Q M. A comparison theorem on moment inequalities between negatively associated and independent random variables[J]. Journal Theoretical Probability, 2000, 13: 343-356.
    [7]
    Liang H Y, Zhang J J. Strong convergence for weighted sums of negatively associated arrays[J]. Chinese Annals of Mathematics, Series B, 2010, 31(2): 273-288.
    [8]
    Wu Q Y, Jiang Y Y. Chovers law of the iterated logarithm for negatively associated sequences[J]. Journal of Systems Science and Complexity, 2010, 23: 293-302.
    [9]
    Wu Q Y, Jiang Y Y. A law of the iterated logarithm of partial sums for negatively associated random variables[J]. Journal of the Korean Statistical Society, 2010, 39: 199-206.
    [10]
    Wang X J, Hu S H, Yang W Z. Some convergence results for arbitrary sequences under moment condition[J]. Chinese Quarterly Journal of Mathematics, 2011, 26 (4): 585-589.
    [11]
    Wang Z Z. On strong law of large numbers for random sequence[J]. Chinese Quarterly Journal of Mathematics, 2010, 25 (4): 475-480.
    [12]
    Shen A T. On strong convergence for weighted sums of a class of random variables[J]. Abstract and Applied Analysis, 2013, 2013: Article ID 216236.
    [13]
    Shen A T. Bernstein-type inequality for widely dependent sequence and its application to nonparametric regression models[J]. Abstract and Applied Analysis, 2013, 2013: Article ID 862602.
    [14]
    Shen A T, Wu R C. Some probability inequalities for a class of random variables and their applications[J]. Journal of Inequalities and Applications, 2013, 2013: 57.
    [15]
    Yang S. Uniformly asymptotic normality of the regression weighted estimator for negatively associated samples[J]. Statistics and Probability Letters, 2003, 2: 101-110.
    [16]
    Matula P. A note on the almost sure convergence of sums of negatively dependent random variables[J]. Statistics and Probability Letters, 1992, 15: 209-213.
    [17]
    吴群英. 混合序列的概率极限理论[M]. 北京: 科学出版社, 2006.
    [18]
    Sung S H. On the strong law of large numbers for pairwise i.i.d. random variables with general moment conditions[J]. Statistics and Probability Letters, 2013, 83: 1 963-1 968.
    [19]
    Wang X J, Li X Q, Hu S H, et al. Strong limit theorems for weighted sums of negatively associated random variables[J]. Stochastic Analysis and Applications, 2011, 29 (1): 1-14.
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