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COMPLETE CONVERGENCE FOR WEIGHTED SUMS OF RANDOM ELEMENTS

  • Sung, Soo-Hak (DEPARTMENT OF APPLIED MATHEMATICS PAI CHAI UNIVERSITY)
  • Published : 2010.03.31

Abstract

We obtain a result on complete convergence of weighted sums for arrays of rowwise independent Banach space valued random elements. No assumptions are given on the geometry of the underlying Banach space. The result generalizes the main results of Ahmed et al. [1], Chen et al. [2], and Volodin et al. [14].

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References

  1. S. E. Ahmed, R. Giuliano Antonini, and A. Volodin, On the rate of complete convergencefor weighted sums of arrays of Banach space valued random elements with applicationto moving average processes, Statist. Probab. Lett. 58 (2002), no. 2, 185–194. https://doi.org/10.1016/S0167-7152(02)00126-8
  2. P. Chen, S. H. Sung, and A. I. Volodin, Rate of complete convergence for arrays ofBanach space valued random elements, Siberian Adv. Math. 16 (2006), no. 3, 1–14.
  3. A. de Acosta, Inequalities for B-valued random vectors with applications to the stronglaw of large numbers, Ann. Probab. 9 (1981), no. 1, 157–161. https://doi.org/10.1214/aop/1176994517
  4. 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
  5. T.-C. Hu, D. Li, A. Rosalsky, and A. Volodin, On the rate of complete convergence for weighted sums of arrays of Banach space valued random elements, Teor. Veroyatnost. i Primenen. 47 (2002), no. 3, 533–547; translation in Theory Probab. Appl. 47 (2003), no. 3, 455–468. https://doi.org/10.4213/tvp3691
  6. T.-C. Hu, A. Rosalsky, D. Szynal, and A. Volodin, On complete convergence for arraysof rowwise independent random elements in Banach spaces, Stochastic Anal. Appl. 17(1999), no. 6, 963–992. https://doi.org/10.1080/07362999908809645
  7. A. Kuczmaszewska and D. Szynal, On complete convergence in a Banach space, Internat.J. Math. Math. Sci. 17 (1994), no. 1, 1–14. https://doi.org/10.1155/S0161171294000013
  8. J. Kuelbs and J. Zinn, Some stability results for vector valued random variables, Ann.Probab. 7 (1979), no. 1, 75–84. https://doi.org/10.1214/aop/1176995149
  9. M. Ledoux and M. Talagrand, Probability in Banach Spaces, Springer-Verlag, 1991.
  10. S. H. Sung, Complete convergence for weighted sums of arrays of rowwise independentB-valued random variables, Stochastic Anal. Appl. 15 (1997), no. 2, 255–267. https://doi.org/10.1080/07362999708809474
  11. S. H. Sung, Complete convergence for weighted sums of random variables, Statist. Probab.Lett. 77 (2007), no. 3, 303–311. https://doi.org/10.1016/j.spl.2006.07.010
  12. S. H. Sung, M. Ord´o˜nez Cabrera, and T.-C. Hu, On complete convergence for arrays ofrowwise independent random elements, J. Korean Math. Soc. 44 (2007), no. 2, 467–476. https://doi.org/10.4134/JKMS.2007.44.2.467
  13. S. H. Sung and A. I. Volodin, On the rate of complete convergence for weighted sumsof arrays of random elements, J. Korean Math. Soc. 43 (2006), no. 4, 815–828. https://doi.org/10.4134/JKMS.2006.43.4.815
  14. A. Volodin, R. Giuliano Antonini, and T.-C. Hu, A note on the rate of completeconvergence for weighted sums of arrays of Banach space valued random elements,Lobachevskii J. Math. 15 (2004), 21–33.
  15. X. Wang, M. B. Rao, and X. Yang, Convergence rates on strong laws of large numbersfor arrays of rowwise independent elements, Stochastic Anal. Appl. 11 (1993), no. 1,115–132. https://doi.org/10.1080/07362999308809305