## Abstract and Applied Analysis

### On Complete Convergence for Weighted Sums of ${\rho }^{\mathbf{*}}$-Mixing Random Variables

#### Abstract

We prove the strong law of large numbers for weighted sums ${\sum }_{i=1}^{n}\mathrm{‍}{a}_{ni}{X}_{i}$, which generalizes and improves the corresponding one for independent and identically distributed random variables and $\phi$-mixing random variables. In addition, we present some results on complete convergence for weighted sums of ${\rho }^{\mathbf{*}}$-mixing random variables under some suitable conditions, which generalize the corresponding ones for independent random variables.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 947487, 7 pages.

Dates
First available in Project Euclid: 27 February 2014

https://projecteuclid.org/euclid.aaa/1393512126

Digital Object Identifier
doi:10.1155/2013/947487

Mathematical Reviews number (MathSciNet)
MR3126800

Zentralblatt MATH identifier
07095525

#### Citation

Shen, Aiting; Wang, Xinghui; Zhu, Huayan. On Complete Convergence for Weighted Sums of ${\rho }^{\mathbf{*}}$ -Mixing Random Variables. Abstr. Appl. Anal. 2013 (2013), Article ID 947487, 7 pages. doi:10.1155/2013/947487. https://projecteuclid.org/euclid.aaa/1393512126

#### References

• Z. D. Bai and P. E. Cheng, “Marcinkiewicz strong laws for linear statistics,” Statistics & Probability Letters, vol. 46, no. 2, pp. 105–112, 2000.
• S. H. Sung, “On the strong convergence for weighted sums of random variables,” Statistical Papers, vol. 52, no. 2, pp. 447–454, 2011.
• G.-H. Cai, “Strong laws for weighted sums of NA random var-iables,” Metrika, vol. 68, no. 3, pp. 323–331, 2008.
• 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.
• X.-C. Zhou, C.-C. Tan, and J.-G. Lin, “On the strong laws forweighted sums of ${\rho }^{\boldsymbol{\ast\,\!}}$-mixing random variables,” Journal of Inequalities and Applications, vol. 2011, Article ID 157816, 8 pages, 2011.
• 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.
• 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.
• X. J. Wang, S. H. Hu, W. Z. Yang, and X. H. Wang, “On com-plete 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.
• Q. Y. Wu and P. Y. Chen, “An improved result in almost sure cen-tral limit theorem for self-normalized products of partial sums,” Journal of Inequalities and Applications, vol. 2013, p. 129, 2013.
• X. F. Tang, “Strong convergence results for arrays of rowwise pairwise NQD random variables,” Journal of Inequalities and Applications, vol. 2013, p. 102, 2013.
• 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.
• X. J. Wang, S. H. Hu, W. Z. Yang, and Y. Shen, “On complete con-vergence for weighed sums of $\varphi$-mixing random variables,” Journal of Inequalities and Applications, vol. 2010, Article ID 372390, 13 pages, 2010.
• 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.
• 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.
• M. Peligrad and A. Gut, “Almost-sure results for a class of dep-endent random variables,” Journal of Theoretical Probability, vol. 12, no. 1, pp. 87–104, 1999.
• 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.
• S. X. Gan, “Almost sure convergence for $\widetilde{\rho }$-mixing random var-iable sequences,” Statistics & Probability Letters, vol. 67, no. 4, pp. 289–298, 2004.
• A. Kuczmaszewska, “On Chung-Teicher type strong law of large numbers for ${\rho }^{\ast\,\!}$-mixing random variables,” Discrete Dynamics in Nature and Society, vol. 2008, Article ID 140548, 10 pages, 2008.
• Q. Y. Wu and Y. Y. Jiang, “Some strong limit theorems for $\widetilde{\rho }$-mix-ing sequences of random variables,” Statistics & Probability Letters, vol. 78, no. 8, pp. 1017–1023, 2008.
• 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.
• 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.
• G.-H. Cai, “Strong law of large numbers for ${\rho }^{\boldsymbol{\ast\,\!}}$-mixing sequences with different distributions,” Discrete Dynamics in Nature and Society, vol. 2006, Article ID 27648, 7 pages, 2006.
• A. Kuczmaszewska, “On complete convergence for arrays of rowwise dependent random variables,” Statistics & Probability Letters, vol. 77, no. 11, pp. 1050–1060, 2007.
• M.-H. Zhu, “Strong laws of large numbers for arrays of rowwise ${\rho }^{\boldsymbol{\ast\,\!}}$-mixing random variables,” Discrete Dynamics in Nature and Society, vol. 2007, Article ID 74296, 6 pages, 2007.
• J. An and D. M. Yuan, “Complete convergence of weighted sumsfor ${\rho }^{\ast\,\!}$-mixing sequence of random variables,” Statistics & Probability Letters, vol. 78, no. 12, pp. 1466–1472, 2008.
• X. J. Wang, X. Q. Li, W. Z. Yang, and S. H. Hu, “On complete con-vergence for arrays of rowwise weakly dependent random variables,” Applied Mathematics Letters, vol. 25, no. 11, pp. 1916–1920, 2012.
• S. H. Sung, “Complete convergence for weighted sums of ${\rho }^{\boldsymbol{\ast\,\!}}$-mixing random variables,” Discrete Dynamics in Nature and Society, vol. 2010, Article ID 630608, 13 pages, 2010.
• 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.
• Q. Y. Wu and Y. Y. Jiang, “Some strong limit theorems for weighted product sums of $\widetilde{\rho }$-mixing sequences of random variables,” Journal of Inequalities and Applications, vol. 2009, Article ID 174768, 10 pages, 2009.
• Q. Y. Wu and Y. Y. Jiang, “Chover-type laws of the $k$-iterated logarithm for $\widetilde{\rho }$-mixing sequences of random variables,” Journal of Mathematical Analysis and Applications, vol. 366, no. 2, pp. 435–443, 2010.
• Q. Y. Wu, “Further study strong consistency of $M$ estimator in linear model for $\widetilde{\rho }$-mixing random samples,” Journal of Systems Science & Complexity, vol. 24, no. 5, pp. 969–980, 2011.
• 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 $\widetilde{\rho }$-mixing sequences,” Abstract and Applied Analysis, vol. 2012, Article ID 907286, 12 pages, 2012.
• Y. F. Wu, C. H. Wang, and A. Volodin, “Limiting behavior forarrays of rowwise ${\rho }^{\boldsymbol{\ast\,\!}}$-mixing random variables,” Lithuanian Mathematical Journal, vol. 52, no. 2, pp. 214–221, 2012.
• M. L. Guo and D. J. Zhu, “Equivalent conditions of complete moment convergence of weighted sums for ${\rho }^{\boldsymbol{\ast\,\!}}$-mixing sequence of random variables,” Statistics & Probability Letters, vol. 83, no. 1, pp. 13–20, 2013.
• Q. Y. Wu, Probability Limit Theory for Mixing Sequences, Science Press of China, Beijing, China, 2006.
• 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.
• 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.
• 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.
• 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.
• 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.
• A. Kuczmaszewska, “On complete convergence for arrays of rowwise negatively associated random variables,” Statistics & Probability Letters, vol. 79, no. 1, pp. 116–124, 2009. \endinput