DOI QR코드

DOI QR Code

ON THE CONVERGENCE OF SERIES FOR ROWWISE SUMS OF NEGATIVELY SUPERADDITIVE DEPENDENT RANDOM VARIABLES

  • Huang, Haiwu (College of Mathematics and Statistics Hengyang Normal University) ;
  • Zhang, Qingxia (School of Sciences Southwest Petroleum University)
  • Received : 2019.04.12
  • Accepted : 2019.08.23
  • Published : 2020.05.31

Abstract

In the paper, some probability convergence properties of series for rowwise sums of negatively superadditive dependent (NSD) random variables are discussed. We establish some sharp results on these convergence for NSD random variables under some general settings, which generalize and improve the corresponding ones of some known literatures.

References

  1. K. Alam and K. M. L. Saxena, Positive dependence in multivariate distributions, Comm. Statist. A-Theory Methods 10 (1981), no. 12, 1183-1196. https://doi.org/10.1080/03610928108828102 https://doi.org/10.1080/03610928108828102
  2. Y. S. Chow, On the rate of moment convergence of sample sums and extremes, Bull. Inst. Math. Acad. Sinica 16 (1988), no. 3, 177-201.
  3. 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 https://doi.org/10.1016/S0047-259X(03)00064-2
  4. X. Deng, X. J. Wang, Y. Wu, and Y. Ding, Complete moment convergence and complete convergence for weighted sums of NSD random variables, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 110 (2016), no. 1, 97-120. https://doi.org/10.1007/s13398-015-0225-7 https://doi.org/10.1007/s13398-015-0225-7
  5. 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 https://doi.org/10.1016/j.spl.2009.12.014
  6. 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), no. 8, 1112-1120. https://doi.org/10.1016/j.spl.2011.03.005 https://doi.org/10.1016/j.spl.2011.03.005
  7. S. Gan and P. Chen, On the limiting behavior of the maximum partial sums for arrays of rowwise NA random variables, Acta Math. Sci. Ser. B (Engl. Ed.) 27 (2007), no. 2, 283-290. https://doi.org/10.1016/S0252-9602(07)60027-7
  8. P. L. Hsu and H. Robbins, Complete convergence and the law of large numbers, Proc. Nat. Acad. Sci. U.S.A. 33 (1947), 25-31. https://doi.org/10.1073/pnas.33.2.25 https://doi.org/10.1073/pnas.33.2.25
  9. T.-C. Hu, Negatively superadditive dependence of random variables with applications, Chinese J. Appl. Probab. Statist. 16 (2000), no. 2, 133-144.
  10. T.-C. Hu and R. L. Taylor, On the strong law for arrays and for the bootstrap mean and variance, Internat. J. Math. Math. Sci. 20 (1997), no. 2, 375-382. https://doi.org/10.1155/S0161171297000483 https://doi.org/10.1155/S0161171297000483
  11. 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 https://doi.org/10.1214/aos/1176346079
  12. J. H. B. Kemperman, On the FKG-inequality for measures on a partially ordered space, Nederl. Akad. Wetensch. Proc. Ser. A 80=Indag. Math. 39 (1977), no. 4, 313-331.
  13. B. Meng, D. Wang, and Q. Wu, Complete convergence and complete moment convergence for arrays of rowwise negatively superadditive dependent random variables, Comm. Statist. Theory Methods 47 (2018), no. 16, 3910-3922. https://doi.org/10.1080/03610926.2017.1364391 https://doi.org/10.1080/03610926.2017.1364391
  14. H. Naderi, M. Amini, and A. Bozorgnia, On the rate of complete convergence for weighted sums of NSD random variables and an application, Appl. Math. J. Chinese Univ. Ser. B 32 (2017), no. 3, 270-280. https://doi.org/10.1007/s11766-017-3437-0 https://doi.org/10.1007/s11766-017-3437-0
  15. Y. Shen, X. Wang, W. Yang, and S. Hu, Almost sure convergence theorem and strong stability for weighted sums of NSD random variables, Acta Math. Sin. (Engl. Ser.) 29 (2013), no. 4, 743-756. https://doi.org/10.1007/s10114-012-1723-6 https://doi.org/10.1007/s10114-012-1723-6
  16. A. Shen, M. Xue, and A. Volodin, Complete moment convergence for arrays of rowwise NSD random variables, Stochastics 88 (2016), no. 4, 606-621. https://doi.org/10.1080/17442508.2015.1110153 https://doi.org/10.1080/17442508.2015.1110153
  17. A. 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 https://doi.org/10.1007/s00184-014-0503-y
  18. X. Wang, X. Deng, L. Zheng, and S. 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 https://doi.org/10.1080/02331888.2013.800066
  19. X.Wang, A. Shen, Z. Chen, and S. 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 https://doi.org/10.1007/s11749-014-0402-6
  20. X. Wang, Y. Wu, and S. Hu, Strong and weak consistency of LS estimators in the EV regression model with negatively superadditive-dependent errors, AStA Adv. Stat. Anal. 102 (2018), no. 1, 41-65. https://doi.org/10.1007/s10182-016-0286-8 https://doi.org/10.1007/s10182-016-0286-8
  21. Y. Wu, On complete moment convergence for arrays of rowwise negatively associated random variables, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 108 (2014), no. 2, 669-681. https://doi.org/10.1007/s13398-013-0133-7 https://doi.org/10.1007/s13398-013-0133-7
  22. Y. Wu, X. Wang, and S. Hu, Complete convergence for arrays of rowwise negatively superadditive-dependent random variables and its applications, Appl. Math. J. Chinese Univ. Ser. B 31 (2016), no. 4, 439-457. https://doi.org/10.1007/s11766-016-3406-z https://doi.org/10.1007/s11766-016-3406-z
  23. Y. Wu and D. Zhu, Convergence properties of partial sums for arrays of rowwise negatively orthant dependent random variables, J. Korean Statist. Soc. 39 (2010), no. 2, 189-197. https://doi.org/10.1016/j.jkss.2009.05.003 https://doi.org/10.1016/j.jkss.2009.05.003