Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

STRONG STABILITY OF A TYPE OF JAMISON WEIGHTED SUMS FOR END RANDOM VARIABLES

  • Yan, Jigao
  • Received : 2016.04.27
  • Published : 2017.05.01

Abstract

In this paper, we consider the strong stability of a type of Jamison weighted sums, which not only extend the corresponding result of Jamison etc. [13] from i.i.d. case to END random variables, but also obtain the necessary and sufficient results. As an important consequence, we present the result of SLLN as that of i.i.d. case.

Keywords

extended negatively dependent;weighted sums;strong stability

File

Download PDF

References

  1. N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular Variation, Cambridge University Press, 1987.
  2. T. Birkel, A note on the strong law of large numbers for positively dependent random variables, Statist. Probab. Lett. 7 (1988), no. 1, 17-20. https://doi.org/10.1016/0167-7152(88)90080-6
  3. H. W. Block, T. H. Savits, and M. Shaked, Some concepts of negative dependence. Ann. Probab. 10 (1982), no. 3, 765-772. https://doi.org/10.1214/aop/1176993784
  4. T. K. Chandra, Uniform integrability in the Cesaro sence and the weak law of large numbers, Sankhya Ser. A 51 (1989), no. 3, 309-317.
  5. Y. Chen, A. Chen, and K. Ng, The strong law of large numbers for extended negatively dependent random variables, J. Appl. Probab. 47 (2010), no. 4, 908-922. https://doi.org/10.1239/jap/1294170508
  6. N. Ebrahimi and M. Ghosh, Multivariate negative dependence, Comm. Statist. A - Theory Methods 10 (1981), no. 4, 307-337. https://doi.org/10.1080/03610928108828041
  7. N. Etemadi, An elementary proof of the strong law of large numbers, Z. Wahrsch. Verw. Gebiete 55 (1981), no. 1, 119-122. https://doi.org/10.1007/BF01013465
  8. N. Etemadi, On the strong law of large numbers for nonnegative random variables, J. Multivariate Anal. 13 (1983), 187-193. https://doi.org/10.1016/0047-259X(83)90013-1
  9. S. Gan, Convergence and laws of large numbers for weighted sums of arrays of Banach valued random elements, J. Wuhan Univ. Natur. Sci. Ed. 43 (1997), no. 5, 569-574.
  10. C. C. Heyde, A supplement to the strong law of large numbers, J. Appl. Probab. 12 (1975), 173-175. https://doi.org/10.2307/3212424
  11. P. L. Hsu and H. Robbins, Complete convergence and the law of large numbers, Proc. Natl. Acad. Sci. USA 33 (1947), 25-31. https://doi.org/10.1073/pnas.33.2.25
  12. T. C. Hu, A. Rosalsky, and K. L. Wang, Complete convergence theorems for extended negatively dependent random variables, Sankhya A 77 (2015), no. 1, 1-29.
  13. B. Jamison, S. Orey, and W. E. Pruitt, Convergence of weighted averages of independent random variables, Z. Wahrsch. Verw. Gebiete 4 (1965), 40-44. https://doi.org/10.1007/BF00535481
  14. L. Liu, Precise large deviations for dependent random variables with heavy tails, Statist. Probab. Lett. 79 (2009), no. 9, 1290-1298. https://doi.org/10.1016/j.spl.2009.02.001
  15. L. Liu, Necessary and sufficient conditions for moderate deviations of dependent random variables with heavy tails, Sci. China Ser. A Math. 53 (2010), no. 6, 1421-1434. https://doi.org/10.1007/s11425-010-4012-9
  16. P. Matula, A note on the almost sure convergence of sums of negatively dependent random variables, Statist. Probab. Lett. 15 (1992), no. 3, 209-213. https://doi.org/10.1016/0167-7152(92)90191-7
  17. V. V. Petrov, A note on the Borel-Cantelli lemma, Statist. Probab. Lett. 58 (2002), no. 3, 283-286. https://doi.org/10.1016/S0167-7152(02)00113-X
  18. D. Qiu, P. Chen, R. Antonini, and A. Volodin, On the complete convergence for arrays of rowwise extended negatively dependent random variables, J. Korean Math. Soc. 50 (2013), no. 2, 379-392. https://doi.org/10.4134/JKMS.2013.50.2.379
  19. Q. M. Shao, On the complete convergence for ${\rho}$-mixing sequences of random variables, Acta Math. Sinica 32 (1989), no. 3, 377-393.
  20. A. Shen, Probability inequalities for END sequence and their applications, J. Inequal. Appl. 98 (2011), 1-12.
  21. W. Stout, Almost Sure Convergence, New York, Academic, 1974.
  22. X. Wang, T. Hu, A. Volodin, and S. Hu, Complete convergence for weighted sums and arrrays of rowwise extended negatively dependent random variables, Comm. Statist. Theory Methods 42 (2013), no. 13, 2391-2401. https://doi.org/10.1080/03610926.2011.609321
  23. Y. B.Wang, C. Su, and X. G. Liu, On some limit properties for pairwise NQD sequences, Acta Math. Appl. Sin. 21 (1998), no. 3, 404-414.
  24. Y. Wu and M. Guan, Convergence properties of the partial sums for sequences of END random variables, J. Korean Math. Soc. 49 (2012), no. 6, 1097-1110. https://doi.org/10.4134/JKMS.2012.49.6.1097
  25. J. Yan, Precise asymptotics of U-statistics, Acta. Math. Sinica 50 (2007), no. 3, 517-526.
  26. J. Yan, The uniform convergence and precise asymptotics of generalized stochastic order statistics, J. Math. Anal. Appl. 343 (2008), no. 2, 644-653. https://doi.org/10.1016/j.jmaa.2008.01.072
  27. Y. Yan and J. Liu, Kolmogorov-type inequality and some limit theorems of extended negatively dependent random variables and its applications, Submitted, 2016.
  28. J. Yan, Y.Wang, and F. Cheng, Precise asymptotics for order statistics of a non-random sample and a random sample, J. Sysems Sci. Math. Sci. 26 (2006), no. 2, 237-244.

Cited by

  1. Almost sure convergence for weighted sums of WNOD random variables and its applications to nonparametric regression models 2017, https://doi.org/10.1080/03610926.2017.1364390
  2. On Complete Convergence in Marcinkiewicz-Zygmund Type SLLN for END Random Variables and Its Applications pp.1532-415X, 2018, https://doi.org/10.1080/03610926.2018.1508709
  3. Complete Convergence and Complete Moment Convergence for Maximal Weighted Sums of Extended Negatively Dependent Random Variables vol.34, pp.10, 2018, https://doi.org/10.1007/s10114-018-7133-7

Acknowledgement

Supported by : National Natural Science Foundation of China