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Link to original content: https://doi.org/10.14736/kyb-2016-1-0052
Kybernetika - Article detail

Kybernetika 52 no. 1, 52-65, 2016

Strong Convergence for weighed sums of negatively superadditive dependent random variables

Zhiyong Chen, Haibin Wang, Xuejun Wang and Shuhe HuDOI: 10.14736/kyb-2016-1-0052

Abstract:

In this paper, the strong law of large numbers for weighted sums of negatively superadditive dependent (NSD, in short) random variables is obtained, which generalizes and improves the corresponding one of Bai and Cheng (\cite{2}) for independent and identically distributed random variables to the case of NSD random variables.

Keywords:

strong law of large numbers, weighted sums, NSD random variables

Classification:

60F15

References:

  1. K. Alam and K. M. L. Saxena: Positive dependence in multivariate distributions. Commun. Statist. - Theory and Methods 10 (1981), 12, 1183-1196.   DOI:10.1080/03610928108828102
  2. Z. D. Bai and P. E. Cheng: Marcinkiewicz strong laws for linear statistics. Statist. Probab. Lett. 46 (2000), 2, 105-112.   DOI:10.1016/s0167-7152(99)00093-0
  3. H. M. Block, T. H. Savits and M. Shaked: Some concepts of negative dependence. Ann. Probab. 10 (1982), 3, 765-772.   DOI:10.1214/aop/1176993784
  4. K. Budsaba, P. Chen, K. Panishkan and A. Volodin: Strong laws for weighted sums and certain types U-statistics based on negatively associated random variables. Siberian Advances in Mathematics 19 (2009), 4, 225-232.   DOI:10.3103/s1055134409040014
  5. T. C. Christofides and E. Vaggelatou: A connection between supermodular ordering and positive/negative association. J. Multivariate Anal. 88 (2004), 1, 138-151.   DOI:10.1016/s0047-259x(03)00064-2
  6. N. Eghbal, M. Amini and A. Bozorgnia: Some maximal inequalities for quadratic forms of negative superadditive dependence random variables. Statist. Probab. Lett. 80 (2010), 7, 587-591.   DOI:10.1016/j.spl.2009.12.014
  7. N. Eghbal, M. Amini and A. Bozorgnia: On the Kolmogorov inequalities for quadratic forms of dependent uniformly bounded random variables. Statist. Probab. Lett. 81 (2011), 8, 1112-1120.   DOI:10.1016/j.spl.2011.03.005
  8. M. Gerasimov, V. Kruglov and A. Volodin: On negatively associated random variables. Lobachevskii J. Math. 33 (2012), 1, 47-55.   DOI:10.1134/s1995080212010052
  9. S. H. Hu, X. T. Liu, X. H. Wang and X. T. Li: Strong law of large numbers of partial sums for pairwise NQD sequences. J. Math. Res. Appl. 33 (2013), 1, 111-116.   CrossRef
  10. T. Z. Hu: Negatively superadditive dependence of random variables with applications. Chinese J. App. Probab. Statist. 16 (2000), 133-144.   CrossRef
  11. K. Joag-Dev and F. Proschan: Negative association ofrandom variables with applications. Ann. Statist. 11 (1983), 1, 286-295.   DOI:10.1214/aos/1176346079
  12. B. Y. Jing and H. Y. Liang: Strong limit theorems for weighted sums of negatively associated random variables. J. Theoret. Probab. 21 (2008), 4, 890-909.   DOI:10.1007/s10959-007-0128-4
  13. P. Matula: A note on the almost sure convergence of sums of negatively dependent random variables. Statistics and Probability Letters, 15 (1992), 209-213.   DOI:10.1016/0167-7152(92)90191-7
  14. Y. J. Meng and Z. Y. Lin: Strong laws of large numbers for $\tilde{\rho}$-mixing random variables. J. Math. Anal. Appl. 365 (2010), 711-717.   DOI:10.1016/j.jmaa.2009.12.009
  15. Q. M. Shao: A comparison theorem on moment inequalities between negatively associated and independent random variables. J. Theoret. Probab. 13 (2000), 2, 343-356.   CrossRef
  16. A. T. Shen and R. C. Wu: Strong and weak convergence for asymptotically almost negatively associated random variables. Discrete Dynamics in Nature and Society 2013 (2013), 1-7.   DOI:10.1155/2013/235012
  17. A. T. Shen: On the strong convergence rate for weighted sums of arrays of rowwise negatively orthant dependent random variables. RACSAM 107 (2013), 2, 257-271.   DOI:10.1007/s13398-012-0067-5
  18. A. T. Shen: On strong convergence for weighted sums of a class of random variables. Abstract Appl. Anal. 2013 (2013), 1-7.   DOI:10.1155/2013/216236
  19. A. T. Shen, Y. Zhang and A. Volodin: Applications of the Rosenthal-type inequality for negatively superadditive dependent random variables. Metrika 78 (2015), 295-311.   DOI:10.1007/s00184-014-0503-y
  20. Y. Shen, X. J. Wang, W. Z. Yang and S. H. Hu: Almost sure convergence theorem and strong stability for weighted sums of NSD random variables. Acta Mathematica Sinica, English Series, 29 (2013), 4, 743-756.   DOI:10.1007/s10114-012-1723-6
  21. Y. Shen, X. J. Wang and S. H. Hu: On the strong convergence and some inequalities for negatively superadditive dependent sequences. J. Inequalities Appl. 2013 (2013), 1, 448.   DOI:10.1186/1029-242x-2013-448
  22. S. H. Sung: On the strong law of large numbers for weighted sums of random variables. Computers Math. Appl. 62 (11) (2011), 4277-4287.   DOI:10.1016/j.camwa.2011.10.018
  23. X. J. Wang, X. Q. Li, S. H. Hu and W. Z. Yang: Strong limit theorems for weighted sums of negatively associated random variables. Stochast. Anal. Appl. 29 (2011), 1, 1-14.   DOI:10.1080/07362994.2010.515484
  24. X. J. Wang, S. H. Hu and W. Z. Yang: Complete convergence for arrays of rowwise negatively orthant dependent random variables. RACSAM 106 (2012), 2, 235-245.   DOI:10.1007/s13398-011-0048-0
  25. X.J. Wang, X. Deng, L.L. Zheng and S.H. Hu: Complete convergence for arrays of rowwise negatively superadditive-dependent random variables and its applications. Statistics 48 (4) (2014), 834-850.   DOI:10.1080/02331888.2013.800066
  26. X. J. Wang, A. T. Shen, Z. Y. Chen and S. H. Hu: Complete convergence for weighted sums of NSD random variables and its application in the EV regression model. Test 24 (2015), 166-184.   DOI:10.1007/s11749-014-0402-6
  27. Q. Y. Wu: Probability Limit Theory for Mixing Sequence. Science Press of China, Beijing 2006.   CrossRef
  28. Q. Y. Wu and Y. Y. Jiang: A law of the iterated logarithm of partial sums for NA random variables. J. Korean Statist. Soc. 39 (2010), 199-206.   DOI:10.1016/j.jkss.2009.06.001
  29. Q. Y. Wu and Y. Y. Jiang: Chover's law of the iterated logarithm for negatively associated sequences. J. Systems Sci. Complex. 23 (2010), 293-302.   DOI:10.1007/s11424-010-7258-y
  30. W. Z Yang, S. H. Hu, X. J. Wang and Q. C. Zhang: Berry-Esséen bound of sample quantiles for negatively associated sequence. J. Inequalities Appl. 2011 (2011), 1, 83.   DOI:10.1186/1029-242x-2011-83