Table of Contents Author Guidelines Submit a Manuscript
Journal of Applied Mathematics and Stochastic Analysis
Volume 2006, Article ID 68605, 6 pages
http://dx.doi.org/10.1155/JAMSA/2006/68605

Mean convergence theorem for multidimensional arrays of random elements in Banach spaces

Department of Mathematics, Vinh University, Nghe An 42118, Vietnam

Received 2 March 2006; Revised 29 June 2006; Accepted 12 July 2006

Copyright © 2006 Le Van Thanh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. A. Adler, M. O. Cabrera, A. Rosalsky, and A. I. Volodin, “Degenerate weak convergence of row sums for arrays of random elements in stable type p Banach spaces,” Bulletin of the Institute of Mathematics. Academia Sinica, vol. 27, no. 3, pp. 187–212, 1999. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. A. Adler, A. Rosalsky, and A. I. Volodin, “A mean convergence theorem and weak law for arrays of random elements in martingale type p Banach spaces,” Statistics & Probability Letters, vol. 32, no. 2, pp. 167–174, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. M. O. Cabrera and A. I. Volodin, “Mean convergence theorems and weak laws of large numbers for weighted sums of random variables under a condition of weighted integrability,” Journal of Mathematical Analysis and Applications, vol. 305, no. 2, pp. 644–658, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. A. de Acosta, “Inequalities for B-valued random vectors with applications to the strong law of large numbers,” The Annals of Probability, vol. 9, no. 1, pp. 157–161, 1981. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. A. Gut, “Marcinkiewicz laws and convergence rates in the law of large numbers for random variables with multidimensional indices,” The Annals of Probability, vol. 6, no. 3, pp. 469–482, 1978. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. V. Mandrekar and J. Zinn, “Central limit problem for symmetric case: convergence to non-Gaussian laws,” Studia Mathematica, vol. 67, no. 3, pp. 279–296, 1980. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. M. B. Marcus and W. A. Woyczyński, “Stable measures and central limit theorems in spaces of stable type,” Transactions of the American Mathematical Society, vol. 251, pp. 71–102, 1979. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. G. Pisier, “Probabilistic methods in the geometry of Banach spaces,” in Probability and Analysis (Varenna, 1985), vol. 1206 of Lecture Notes in Mathematics, pp. 167–241, Springer, Berlin, 1986. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. A. Rosalsky and M. Sreehari, “Lr convergence theorems with or without random indices for randomly weighted sums of random elements in martingale type p Banach spaces,” Stochastic Modelling and Applications, vol. 4, pp. 1–15, 2001. View at Google Scholar
  10. A. Rosalsky, M. Sreehari, and A. I. Volodin, “Mean convergence theorems with or without random indices for randomly weighted sums of random elements in Rademacher type p Banach spaces,” Stochastic Analysis and Applications, vol. 21, no. 5, pp. 1169–1187, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. L. V. Thanh, “On the Lp-convergence for multidimensional arrays of random variables,” International Journal of Mathematics and Mathematical Sciences, vol. 2005, no. 8, pp. 1317–1320, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. D. Wei and R. L. Taylor, “Convergence of weighted sums of tight random elements,” Journal of Multivariate Analysis, vol. 8, no. 2, pp. 282–294, 1978. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. W. A. Woyczyński, “Geometry and martingales in Banach spaces. II: independent increments,” in Probability on Banach Spaces, J. Kuelbs and P. Ney, Eds., vol. 4 of Advances in Probability and Related Topics, pp. 267–517, Marcel Dekker, New York, 1978. View at Google Scholar · View at MathSciNet