&rho|混合随机变量加权和的收敛性

数学物理学报 ›› 2011, Vol. 31 ›› Issue (1): 132-141.

&rho|混合随机变量加权和的收敛性

邱德华

广东商学院数学与计算科学学院广州 |510320

收稿日期:2008-12-04 修回日期:2009-10-26 出版日期:2011-02-25 发布日期:2011-02-25
基金资助:
广东省自然科学基金(8151032001000006)资助

Convergence Properties for Weighted Sums of ρ -mixing Random Variables

QIU De-Hua

School of Mathematics and Computational Science, Guangdong University of Business Studies, Guangzhou 510320

Received:2008-12-04 Revised:2009-10-26 Online:2011-02-25 Published:2011-02-25
Supported by:
广东省自然科学基金(8151032001000006)资助

摘要/Abstract

摘要：

该文研究了ρ 混合随机变量加权和的强大数律及完全收敛性, 获得了一些新的结果. 该文的结果推广和改进了Bai 等^[1]及Baum 等^[18] 在 i.i.d. 情形时相应的结果, 也推广和改进了Volodin 等^[4]在实值独立时相应的结果. 该文还得到了一关于任意随机变量阵列加权和的完全收敛性定理.

关键词: ρ 混合随机变量, 加权和, 收敛性

Abstract:

In this paper, the author studies strong law of large numbers and complete convergence for weighted sums of ρ-mixing random variables. The author obtains some new results. The results extend and improve the corresponding results of Bai et al^[1] and Baum et al^[18] from i.i.d. case to ρ-mixing setting. Meanwhile, the results extend and improve the corresponding results of Volodin et al^[4] from real-valued independent random variables to ρ-mixing setting. Furthermore, a complete convergence theorem for weighted sums of arrays of arbitrary random variables is obtained.

Key words: ρ-mixing random variables, Weighted sums, Convergence properties

中图分类号:

60F15

邱德华. &rho|混合随机变量加权和的收敛性[J]. 数学物理学报, 2011, 31(1): 132-141.

QIU De-Hua. Convergence Properties for Weighted Sums of ρ -mixing Random Variables[J]. Acta mathematica scientia,Series A, 2011, 31(1): 132-141.

参考文献

[1] Bai Z D, Cheng P E. Marcinkiewicz strong laws for linear statistics. Statist Probab Lett, 2000, 46: 105--112

[2] Hu T C, Li D, Rosalsky A, et al. On the rate of complete convergence for weighted sums of arrays of Banach space valued random elements. Theory Probab Appl, 2002, 47(3): 455--468

[3] Ahmed S E, Giuliano Antonini R, Volodin A. On the rate of complete convergence for weighted sums of arrays of Banach space valued
random elements with application to moving average processes. Statist Probab Lett, 2002, 58: 185--194

[4] Volodin A, Giuliano Antonini R, Hu T C. A note on the rate of complete convergence for weighted sums of arrays of Banach space valued random elements. Lobachevskii J Math, 2004, 15: 21--33

[5] Stein C. A Bound on the Error in the Normal Approximation to the Distribution of a sum of Dependent Random Variables. Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability. Berkeley: University of California Press, 1972: 583--602

[6] Bryc W, Smolenski W. Moment conditions for almost sure convergence of weakly correlated random variables. Proc Amer Math Soc, 1993, 119: 629--635

[7] Bradley R C. On the spectral density and asymptotic normality of weakly dependent random fields. J Theor Probab, 1992, 5: 355--373

[8] Bradley R C. Equivalent mixing conditions for random fields. Ann Probab, 1993, 21: 1921--1926

[9] 杨善朝. 一类随机变量部分和的矩不等式及其应用.科学通报, 1998, 43(17): 1823--1827

[10] Peligrad M, Gut A. Almost sure results for a class of dependent random variables. J Theoret Probab, 1999, 12: 87--104

[11] 吴群英. ρ 混合序列的若干收敛性质.工程数数学学报, 2001, 18(3): 58--64

[12] 吴群英. ρ 混合序列加权和的完全收敛性和强收敛性.应用数学, 2002, 15(1): 1--4

[13] Gan Shixin. Almost sure convergence for ρ-mixing random variable sequences. Statist Probab Lett, 2004, 67: 289--298

[14] Utev S, Peligrad M. Maximal inequalities and an invariance principle for a class of weakly dependent random variables. J Theoret Probab, 2003, 16: 101--115

[15] 陈平炎. 同分布ρ 混合序列的最大值不等式及其应用研究.数学物理学报, 2006, 26(5): 725--730

[16] 邱德华, 甘师信. ρ 混合阵列的收敛性.数学物理学报, 2005, 25A(1): 73--78

[17] 邱德华, 甘师信. ρ 混合序列的Hájeck-Rènyi型不等式及强大数律.数学的实践与认识, 2007, 37(3): 107--111

[18] Baum L E, Katz M. Convergence rates in the law of large numbers. Transactions of the American Mathematics Society, 1965, 120: 108--123