DOI QR코드

DOI QR Code

ON THE STRONG LAW OF LARGE NUMBERS FOR WEIGHTED SUMS OF NEGATIVELY SUPERADDITIVE DEPENDENT RANDOM VARIABLES

  • SHEN, AITING (SCHOOL OF MATHEMATICAL SCIENCES ANHUI UNIVERSITY)
  • 투고 : 2014.04.14
  • 발행 : 2016.01.01

초록

Let {$X_n,n{\geq}1$} be a sequence of negatively superadditive dependent random variables. In the paper, we study the strong law of large numbers for general weighted sums ${\frac{1}{g(n)}}{\sum_{i=1}^{n}}{\frac{X_i}{h(i)}}$ of negatively superadditive dependent random variables with non-identical distribution. Some sufficient conditions for the strong law of large numbers are provided. As applications, the Kolmogorov strong law of large numbers and Marcinkiewicz-Zygmund strong law of large numbers for negatively superadditive dependent random variables are obtained. Our results generalize the corresponding ones for independent random variables and negatively associated random variables.

키워드

과제정보

연구 과제 주관 기관 : National Natural Science Foundation of China

참고문헌

  1. T. C. Christofides and E. Vaggelatou, A connection between supermodular ordering and positive/negative association, J. Multivariate Anal. 88 (2004), no. 1, 138-151. https://doi.org/10.1016/S0047-259X(03)00064-2
  2. N. Eghbal, M. Amini, and A. Bozorgnia, Some maximal inequalities for quadratic forms of negative superadditive dependence random variables, Statist. Probab. Lett. 80 (2010), no. 7-8, 587-591. https://doi.org/10.1016/j.spl.2009.12.014
  3. N. Eghbal, On the Kolmogorov inequalities for quadratic forms of dependent uniformly bounded random variables, Statist. Probab. Lett. 81 (2011), no. 8, 1112-1120. https://doi.org/10.1016/j.spl.2011.03.005
  4. T. Z. Hu, Negatively superadditive dependence of random variables with applications, Chinese J. Appl. Probab. Statist. 16 (2000), no. 2, 133-144.
  5. R. Jajte, On the strong law of large numbers, Ann. Probab. 31 (2003), no. 1, 409-412. https://doi.org/10.1214/aop/1046294315
  6. B. Y. Jing and H. Y. Liang, Strong limit theorems for weighted sums of negatively associated random variables, J. Theoret. Probab. 21 (2008), no. 4, 890-909. https://doi.org/10.1007/s10959-007-0128-4
  7. K. Joag-Dev and F. Proschan, Negative association of random variables with applications, Ann. Statist. 11 (1983), no. 1, 286-295. https://doi.org/10.1214/aos/1176346079
  8. J. H. B. Kemperman, On the FKG-inequalities for measures on a partially ordered space, Nederl. Akad. Wetensch. Proc. Ser. A 80 (1977), no. 4, 313-331.
  9. Y. J. Meng and Z. Y. Lin Strong laws of large numbers for $\rho$-mixing random variables, J. Math. Anal. Appl. 365 (2010), no. 2, 711-717. https://doi.org/10.1016/j.jmaa.2009.12.009
  10. A. T. Shen, Y. Zhang, and A. Volodin, Applications of the Rosenthal-type inequality for negatively superadditive dependent random variables, Metrika 78 (2015), no. 3, 295-311. https://doi.org/10.1007/s00184-014-0503-y
  11. 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 Math. Sin. English Series 29 (2012), no. 4, 743-756. https://doi.org/10.1007/s10114-012-1723-6
  12. S. H. Sung, On the strong law of large numbers for weighted sums of random variables, Comput. Math. Appl. 62 (2011), no. 11, 4277-4287. https://doi.org/10.1016/j.camwa.2011.10.018
  13. X. F. Tang, Some strong laws of large numbers for weighted sums of asymptotically almost negatively associated random variables, J. Inequal. Appl. 2013 (2013), Article ID 4, 11 pages. https://doi.org/10.1186/1029-242X-2013-11
  14. X. H. Wang and S. H. Hu, On the strong consistency of M-estimates in linear models for negatively superadditive dependent errors, Aust. New Zealand J. Stat. 57 (2015), no. 2, 259-274. https://doi.org/10.1111/anzs.12117
  15. 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 (2014), no. 4, 834-850. https://doi.org/10.1080/02331888.2013.800066
  16. X. J. Wang, S. H. Hu, A. Shen, and W. Z. Yang, An exponential inequality for a NOD sequence and a strong law of large numbers, Appl. Math. Lett. 24 (2011), no. 2, 219-223. https://doi.org/10.1016/j.aml.2010.09.007
  17. X. J. Wang, S. H. Hu, and W. Z. Yang, Complete convergence for arrays of rowwise negatively orthant dependent random variables, RACSAM 106 (2012), no. 2, 235-245. https://doi.org/10.1007/s13398-011-0048-0
  18. 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), no. 1, 166-184. https://doi.org/10.1007/s11749-014-0402-6
  19. X. J. Wang, C. Xu, T.-C. Hu, A. Volodin, and S. H. Hu, On complete convergence for widely orthant-dependent random variables and its applications in nonparametric regression models, TEST 23 (2014), no. 3, 607-629. https://doi.org/10.1007/s11749-014-0365-7
  20. Z. Z. Wang, On strong law of large numbers for dependent random variables, J. Inequal. Appl. 2011 (2011), Article ID 279754, 13 pages. https://doi.org/10.1186/1029-242X-2011-13
  21. Q. Y. Wu, A complete convergence theorem for weighted sums of arrays of rowwise negatively dependent random variables, J. Inequal. Appl. 2012 (2012), Article ID 50, 10 pages. https://doi.org/10.1186/1029-242X-2012-10
  22. Q. Y. Wu and Y. Y. Jiang, The strong consistency of M estimator in a linear model for negatively dependent random samples, Comm. Statist. Theory Methods 40 (2011), no. 3, 467-491. https://doi.org/10.1080/03610920903427792