JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2016, Vol. 51 ›› Issue (2): 50-57.doi: 10.6040/j.issn.1671-9352.0.2015.106

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Almost sure central limit theorem for self-normalized products of some partial sums of NA random variables

XU Feng, WU Qun-ying*   

  1. College of Science, Guilin University of Technology, Guilin 541004, Guangxi, China
  • Received:2015-03-12 Online:2016-02-16 Published:2016-03-11

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Abstract: Let {X,Xn}n∈N be a stationary sequence of NA positive random variables. We proved an almost sure central limit theorem for the self-normalized products of some partial sums(∏ki=1(Sk,i/((k-1)μ)))μ/(βVk), where β>0 was a constant, E(X)=μ, Sk,i=∑kj=1Xj-Xi, 1≤i≤k, V2k=∑ki=1(Xi-μ)2. The results generalize not only on the weigh of the almost sure central limit theorem but also in the range of random variables.

Key words: self-normalized, products of sums, almost sure central limit theorem, NA sequences

CLC Number: 

  • O211
[1] JOAG-DEV K, PROSCHAN F. Negative association of random variables with applications[J]. Ann Statist, 1983, 11:286-295.
[2] ARNOLD B C, VILLASE(~overN)R J A. The asymptotic distribution of sums of records[J].Extremes, 1998, 1(3):351-363.
[3] PANG Tianxiao, LIN Zhangyan, HWANG K S. Asymptotics for self-normalized random products of sums of i.i.d. random variables[J]. Math Anal Appl, 2007, 334:1246-1259.
[4] ZHANG Yong, YANG Xiaoyun. An almost sure central limit theorem for self-normalized products of sums of i.i.d. random variables[J]. Math Anal Appl, 2011, 376:29-41.
[5] WU Qunying, CHEN Pingyan. An improved result in almost sure central limit theorem for self-normalized products of partial sums[J]. Journal of Inequalities and Applications, 2013, 2013:129.1-129.12.
[6] YU Miao. Central limit theorem and almost sure central limit theorem for the product of some partial sums[J]. Proc Indian Acad Sci Math Sci, 2008, 118(2):289-294.
[7] NEWMAN C M. Asymptotic independence and limit theorems for positively and negatively dependent variables[J]. Statist Probab Lett, 1984, 5:127-140.
[8] SU Chun, ZHAO Lincheng, WANG Yuebao. Moment inequalities and weak convergence for negatively associated sequences[J]. Science in China Series A, 1997, 40(2):172-182.
[9] BILLINGSLEY P. Convergence of probability measures [M]. New York:Wiley, 1968.
[10] PELIGRAD M, SHAO Qiman. A note on the almost sure central limit theorem for weakly dependent random variables[J]. Statistics and Probability Letters, 1995, 22:131-136.
[11] ZHANG Lixin. The weak convergence for functions of negatively associated random variables[J]. Multivariate Anal, 2001, 78:272-298.
[12] 苏淳,王岳宝.同分布NA序列的强收敛性[J].应用概率统计,1998,14(2):131-140. SU Chun, WANG Yuebao. Strong convergence for IDNA sequences[J].Chinese Journal of Applied Probability and Statistics, 1998, 14(2):131-140.
[13] 吴群英.混合序列的概率极限理论[M].北京:科学出版社,2006:17. WU Qunying. Probability limit theorems of maxing sequences [M]. Beijing: Science Press, 2006:17.
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