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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 947487, 7 pages
http://dx.doi.org/10.1155/2013/947487
Research Article

On Complete Convergence for Weighted Sums of -Mixing Random Variables

School of Mathematical Science, Anhui University, Hefei 230039, China

Received 29 July 2013; Revised 15 September 2013; Accepted 17 September 2013

Academic Editor: Ciprian A. Tudor

Copyright © 2013 Aiting Shen et al. 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. Z. D. Bai and P. E. Cheng, “Marcinkiewicz strong laws for linear statistics,” Statistics & Probability Letters, vol. 46, no. 2, pp. 105–112, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. S. H. Sung, “On the strong convergence for weighted sums of random variables,” Statistical Papers, vol. 52, no. 2, pp. 447–454, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. G.-H. Cai, “Strong laws for weighted sums of NA random variables,” Metrika, vol. 68, no. 3, pp. 323–331, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. B.-Y. Jing and H.-Y. Liang, “Strong limit theorems for weighted sums of negatively associated random variables,” Journal of Theoretical Probability, vol. 21, no. 4, pp. 890–909, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. X.-C. Zhou, C.-C. Tan, and J.-G. Lin, “On the strong laws for weighted sums of ρ*-mixing random variables,” Journal of Inequalities and Applications, vol. 2011, Article ID 157816, 8 pages, 2011. View at Publisher · View at Google Scholar · View at Scopus
  6. X. J. Wang, S. H. Hu, and A. I. Volodin, “Strong limit theorems for weighted sums of NOD sequence and exponential inequalities,” Bulletin of the Korean Mathematical Society, vol. 48, no. 5, pp. 923–938, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. X. J. Wang, S. H. Hu, and W. Z. Yang, “Complete convergence for arrays of rowwise negatively orthant dependent random variables,” Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales A, vol. 106, no. 2, pp. 235–245, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. X. J. Wang, S. H. Hu, W. Z. Yang, and X. H. Wang, “On complete convergence of weighted sums for arrays of rowwise asymptotically almost negatively associated random variables,” Abstract and Applied Analysis, vol. 2012, Article ID 315138, 15 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. Q. Y. Wu and P. Y. Chen, “An improved result in almost sure central limit theorem for self-normalized products of partial sums,” Journal of Inequalities and Applications, vol. 2013, p. 129, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  10. X. F. Tang, “Strong convergence results for arrays of rowwise pairwise NQD random variables,” Journal of Inequalities and Applications, vol. 2013, p. 102, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  11. G.-H. Cai, “Strong laws for weighted sums of i.i.d. random variables,” Communications of the Korean Mathematical Societ, vol. 21, no. 4, pp. 771–778, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. X. J. Wang, S. H. Hu, W. Z. Yang, and Y. Shen, “On complete convergence for weighed sums of φ-mixing random variables,” Journal of Inequalities and Applications, vol. 2010, Article ID 372390, 13 pages, 2010. View at MathSciNet
  13. R. C. Bradley, “On the spectral density and asymptotic normality of weakly dependent random fields,” Journal of Theoretical Probability, vol. 5, no. 2, pp. 355–373, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. W. Bryc and W. Smoleński, “Moment conditions for almost sure convergence of weakly correlated random variables,” Proceedings of the American Mathematical Society, vol. 119, no. 2, pp. 629–635, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. M. Peligrad and A. Gut, “Almost-sure results for a class of dependent random variables,” Journal of Theoretical Probability, vol. 12, no. 1, pp. 87–104, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. S. Utev and M. Peligrad, “Maximal inequalities and an invariance principle for a class of weakly dependent random variables,” Journal of Theoretical Probability, vol. 16, no. 1, pp. 101–115, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. S. X. Gan, “Almost sure convergence for ρ˜-mixing random variable sequences,” Statistics & Probability Letters, vol. 67, no. 4, pp. 289–298, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  18. A. Kuczmaszewska, “On Chung-Teicher type strong law of large numbers for ρ*-mixing random variables,” Discrete Dynamics in Nature and Society, vol. 2008, Article ID 140548, 10 pages, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  19. Q. Y. Wu and Y. Y. Jiang, “Some strong limit theorems for ρ˜-mixing sequences of random variables,” Statistics & Probability Letters, vol. 78, no. 8, pp. 1017–1023, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  20. X. J. Wang, S. H. Hu, Y. Shen, and N. X. Ling, “Strong law of large numbers and growth rate for a class of random variable sequence,” Statistics & Probability Letters, vol. 78, no. 18, pp. 3330–3337, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  21. X. J. Wang, S. H. Hu, Y. Shen, and W. Z. Yang, “Some new results for weakly dependent random variable sequences,” Chinese Journal of Applied Probability and Statistics, vol. 26, no. 6, pp. 637–648, 2010. View at Zentralblatt MATH · View at MathSciNet
  22. G.-H. Cai, “Strong law of large numbers for ρ*-mixing sequences with different distributions,” Discrete Dynamics in Nature and Society, vol. 2006, Article ID 27648, 7 pages, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  23. A. Kuczmaszewska, “On complete convergence for arrays of rowwise dependent random variables,” Statistics & Probability Letters, vol. 77, no. 11, pp. 1050–1060, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. M.-H. Zhu, “Strong laws of large numbers for arrays of rowwise ρ*-mixing random variables,” Discrete Dynamics in Nature and Society, vol. 2007, Article ID 74296, 6 pages, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  25. J. An and D. M. Yuan, “Complete convergence of weighted sums for ρ*-mixing sequence of random variables,” Statistics & Probability Letters, vol. 78, no. 12, pp. 1466–1472, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  26. X. J. Wang, X. Q. Li, W. Z. Yang, and S. H. Hu, “On complete convergence for arrays of rowwise weakly dependent random variables,” Applied Mathematics Letters, vol. 25, no. 11, pp. 1916–1920, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. S. H. Sung, “Complete convergence for weighted sums of ρ*-mixing random variables,” Discrete Dynamics in Nature and Society, vol. 2010, Article ID 630608, 13 pages, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  28. M. Peligrad, “Maximum of partial sums and an invariance principle for a class of weak dependent random variables,” Proceedings of the American Mathematical Society, vol. 126, no. 4, pp. 1181–1189, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. Q. Y. Wu and Y. Y. Jiang, “Some strong limit theorems for weighted product sums of ρ˜-mixing sequences of random variables,” Journal of Inequalities and Applications, vol. 2009, Article ID 174768, 10 pages, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  30. Q. Y. Wu and Y. Y. Jiang, “Chover-type laws of the k-iterated logarithm for ρ˜-mixing sequences of random variables,” Journal of Mathematical Analysis and Applications, vol. 366, no. 2, pp. 435–443, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  31. Q. Y. Wu, “Further study strong consistency of M estimator in linear model for ρ˜-mixing random samples,” Journal of Systems Science & Complexity, vol. 24, no. 5, pp. 969–980, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  32. X. J. Wang, F. X. Xia, M. M. Ge, S. H. Hu, and W. Z. Yang, “Complete consistency of the estimator of nonparametric regression models based on ρ˜-mixing sequences,” Abstract and Applied Analysis, vol. 2012, Article ID 907286, 12 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  33. Y. F. Wu, C. H. Wang, and A. Volodin, “Limiting behavior for arrays of rowwise ρ*-mixing random variables,” Lithuanian Mathematical Journal, vol. 52, no. 2, pp. 214–221, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  34. M. L. Guo and D. J. Zhu, “Equivalent conditions of complete moment convergence of weighted sums for ρ*-mixing sequence of random variables,” Statistics & Probability Letters, vol. 83, no. 1, pp. 13–20, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  35. Q. Y. Wu, Probability Limit Theory for Mixing Sequences, Science Press of China, Beijing, China, 2006.
  36. X. F. Tang, “Some strong laws of large numbers for weighted sums of asymptotically almost negatively associated random variables,” Journal of Inequalities and Applications, vol. 2013, p. 4, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  37. Z. D. Bai and C. Su, “The complete convergence for partial sums of i.i.d. random variables,” Scientia Sinica A, vol. 28, no. 12, pp. 1261–1277, 1985. View at Zentralblatt MATH · View at MathSciNet
  38. Q.-M. Shao, “A comparison theorem on moment inequalities between negatively associated and independent random variables,” Journal of Theoretical Probability, vol. 13, no. 2, pp. 343–356, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  39. 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, 2013. View at Publisher · View at Google Scholar
  40. D. M. Yuan and J. An, “Rosenthal type inequalities for asymptotically almost negatively associated random variables and applications,” Science in China A, vol. 52, no. 9, pp. 1887–1904, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  41. A. Kuczmaszewska, “On complete convergence for arrays of rowwise negatively associated random variables,” Statistics & Probability Letters, vol. 79, no. 1, pp. 116–124, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet