DOI QR코드

DOI QR Code

LIMITING BEHAVIOR OF THE MAXIMUM OF THE PARTIAL SUM FOR NEGATIVELY SUPERADDITIVE DEPENDENT RANDOM VARIABLES

  • Received : 2015.04.17
  • Accepted : 2015.07.22
  • Published : 2015.08.15

Abstract

In this paper, some $L_p$-convergences and complete convergences of the maximum of the partial sum for negatively superadditive dependent random variables are obtained. The proofs of the results are based on a new Rosenthal type inequality concerning negatively superadditive dependent random variables.

Keywords

References

  1. H. W. Block, W. S. Griffths, and T. H. Savits, L-superadditive structure functions, Adv. Appl. Probab. 21 (1989), 919-929. https://doi.org/10.2307/1427774
  2. T. C. Christofides and E. Vaggelatou, A connection between supermodular ordering and positive/negative association, J. Multivar. Anal. 88 (2004), 138-151. https://doi.org/10.1016/S0047-259X(03)00064-2
  3. N. Eghbal, M. Amini, and A. Bozorgnia, Some maximal inequalities for quadratic forms of negative superadditive dependence random variables, Stat. Probab. Lett. 80 (2010), 587-591. https://doi.org/10.1016/j.spl.2009.12.014
  4. N. Eghbal, M. Amini, and A. Bozorgnia, On the Kolmogorov inequalities for quadratic forms of dependent uniformly bounded random variables, Stat. Probab. Lett. 81 (2011), 1112-1120. https://doi.org/10.1016/j.spl.2011.03.005
  5. P. L. Hsu and H. Robbins, Complete convergence and the law of large numbers, Proc. Natl. Acad. Sci. 33 (1947), 25-31. https://doi.org/10.1073/pnas.33.2.25
  6. T. Z. Hu, Negatively superadditive dependence of random variables with applications, Chinese J. Appl. Probab. Statist. 10 (2000), 133-144.
  7. K. Joag-Dev and F. Proschan, Negative association of random variables with applications, Ann. Statist. 11 (1983), 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), 313-331. https://doi.org/10.1016/1385-7258(77)90027-0
  9. D. Landers and L. Rogge, Laws of large numbers for uncorrelated Cesaro uniformly integrable random variables, Sankhya Ser. A. 59 (1997), 301-310.
  10. A. T. Shen, On the convergence rate for weighted sums of arrays of rowwise negatively orthant dependent random variables, RACSAM 107 (2013), no. 2, 257-271. https://doi.org/10.1007/s13398-012-0067-5
  11. Y. Shen, X. J. Wang, X. Z. Yang, and S. H. Hu, Almost sure convergence theorem and strong stability for weighted sums of NSD random variables, Acta Math. Sin. Engl. Ser. 29 (2013 a), 743-756. https://doi.org/10.1007/s10114-012-1723-6
  12. Y. Shen, X. Wang, and S. H. Hu, On the strong convergence and some inequalities for negatively superadditive dependent sequences, J. Inequl. Appl. (2013b).
  13. 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: A Journal of Theoretical and Applied Statistics (2013), DOI:10.1080/02331888.2013.800066, in press.
  14. Q. Y. Wu, Probability limit theorey for mixing sequences, Science Press of China, Beijing, (2006).
  15. D. M. Yuan and X. S. Wu, Limiting behavior of the maximum of the partial sum for asymptotically negatively associated random variables under residual Cesaro alpha-integrability assumption, J. Statist. Plan, Infer. (2010).