Complete Convergence and Weak Law of Large Numbers for ρ-Mixing Sequences of Random Variables

Qunying Wu

College of Science, Guilin University of Technology, Guilin, China.

DOI: 10.4236/ojs.2012.25062                

Abstract

In this paper, the complete convergence and weak law of large numbers are established for ρ-mixing sequences of random variables. Our results extend and improve the Baum and Katz complete convergence theorem and the classical weak law of large numbers, etc. from independent sequences of random variables to ρ-mixing sequences of random variables without necessarily adding any extra conditions.

Keywords

ρ-Mixing Sequence of Random Variables; Complete Convergence; Weak Law of Large Number

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	W. Bryc and W. Smolenski, “Moment Conditions for Almost Sure Convergence of Weakly Correlated Random Variables,” Proceedings of the American Mathematical Society, Vol. 199, No. 2, 1993, pp. 629-635. doi:10.1090/S0002-9939-1993-1149969-7
[2]	R. C. Bradley, “On the Spectral Density and Asymptotic Normality of Weakly Dependent Random Fields,” Journal of Theoretical Probability, Vol. 5, No. 2, 1992, pp. 355-373.
[3]	S. C. Yang, “Some Moment Inequalities for Partial Sums of Random Variables and Their Applications,” Chinese Science Bulletin, Vol. 43, No. 17, 1998, pp. 1823-1827. doi:10.1007/BF02883381
[4]	Q. Y. Wu and Y. Y. Jiang, “Some Strong Limit Theorems for -Mixing Sequences of Random Variables,” Statistics & Probability Letters, Vol. 78, No. 8, 2008, pp. 1017-1023.
[5]	Q. Y. Wu and Y. Y. Jiang, “Some Strong Limit Theorems for Weighted Product Sums of -Mixing Sequences of Random Variables,” Journal of Inequalities and Applications, 2009. doi:10.1155/2009/174768
[6]	Q. Y. Wu and Y. Y. Jiang, “Chover-Type Laws of the K-Iterated Logarithm for -Mixing Sequences of Random Variables,” Journal of Mathematical Analysis and Applications, Vol. 366, No. 2, 2010, pp. 435-443. doi: 10.1016/j.jmaa.2009.
[7]	Q. Y. Wu, “Further Study Strong Consistency of Estimator in Linear Model for -Mixing Random Samples,” Journal of Systems Science and Complexity, Vol. 24, No. 5, 2011, pp. 969-980. doi:10.1007/s11424-011-8407-7
[8]	M. Peligrad and A. Gut, “Almost-Sure Results for a Class of Dependent Random Variables,” Journal of Theoretical Probability, Vol. 12, No. 1, 1999, pp. 87-104.
[9]	S. X. Gan, “Almost Sure Convergence for -Mixing Random Variable Sequences,” Statistics and Probability Letters, Vol. 67, 2004, pp. 289-298.
[10]	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, 2003, pp. 101-115. doi:10.1023/A:1022278404634
[11]	W. Feller, “A Limit Theorem for Random Variables with Infinite Moments,” American Journal of Mathematics, Vol. 68, 1946, pp. 257-262.
[12]	L. E. Baum and M. Katz, “Convergence Rates in the Law of Large Numbers,” Transactions of the American Mathematical Society, Vol. 120, No. 1, 1965, pp. 108-123.
[13]	Z. D. Bai and C. Su, “The Complete Convergence for Partial Sums of iid Random Variables,” Science in China, Series A, Vol. 5, 1985, pp. 309-412.