﻿ 关于任意随机序列加权和的强收敛性 On Almost Sure Convergence for Weighted Sums of Arbitrarily Dependent Random Sequences

Pure Mathematics
Vol.06 No.01(2016), Article ID:16808,7 pages
10.12677/PM.2016.61004

On Almost Sure Convergence for Weighted Sums of Arbitrarily Dependent Random Sequences

Qiong Wang, Ying Cui, Aihua Fan*

School of Mathematics & Physics Science and Engineering, Anhui University of Technology, Ma’anshan Anhui

Received: Dec. 17th, 2015; accepted: Jan. 19th, 2016; published: Jan. 22nd, 2016

ABSTRACT

Let be a sequence of arbitrarily dependent random variables with. In this paper, by using the Borel-Cantelli lemma and the pure analysis method in probability limit theory, some strong convergence of a class of dependent random variables is discussed and some sufficient conditions on strong law of large numbers for weighted sums of arbitrarily random sequences are also obtained. Some classical results are generalized.

Keywords:Randomized Controlled, Weighted Sums, Strong Law of Large Numbers

1. 引言

2. 预备知识

3. 主要结论与证明

(1)

(2)

(3)

, a.s. (4)

.

,

,

, a.s.

, a.s.

,

, a.s.

, a.s.

, , (5)

, a.s.

, a.s.

, a.s. (6)

, a.s.

,

, a.s.

, ,

, (7)

, a.s.

,

,

，易知且有

. (8)

，由(8)得

,

, a.s.

On Almost Sure Convergence for Weighted Sums of Arbitrarily Dependent Random Sequences[J]. 理论数学, 2016, 06(01): 23-29. http://dx.doi.org/10.12677/PM.2016.61004

1. 1. 格涅坚科, 科尔莫哥洛夫. 相互独立随机变数之和的极限分布[M]. 北京: 科学出版社, 1955.

2. 2. 佩特罗夫. 独立随机变量之和的极限定理[M]. 合肥: 中国科学技术大学出版社, 1991.

3. 3. Rosalsky, A. and Stoica, G. (2010) On the Strong Law of Large Numbers for Identically Distributed Random Variables Irrespective of their Joint Distributions. Statistics & Probability Letters, 80,1265-1270. http://dx.doi.org/10.1016/j.spl.2010.04.005

4. 4. 杨卫国, 刘文. 关于任意随机序列的强收敛性[J]. 数学物理学报, 2003, 23(5): 565-572.

5. 5. 汪忠志. 关于M值随机序列的一个普遍成立的强大数定理[J]. 纯粹数学与应用数学, 2004, 20(4): 327-333.

6. 6. Korchevsky, V.M. (2011) On the Strong Law of Large Numbers for Sequences of Random Variables without the Independence Condition. Vestnik St. Petersburg University: Mathematics, 44, 268-271. http://dx.doi.org/10.3103/S1063454111040066

7. 7. 汪忠志, 徐付霞. 关于B值随机元序列的强收敛性[J]. 纯粹数学与应用数学, 2002, 18(2): 187-190.

8. 8. 张丽娜. 任意B值随机变量序列的强收敛性[J]. 数学杂志, 2002, 22(3): 297-300.

9. 9. Sung, S.H. (1998) SLIN for Weighted Sums of Stochastically Dominated Pairwish Independent Random Variables. Communications of the Korean Mathematical Society, 13, 377-384.

10. 10. 刘京军, 甘师信. 随机变量序列加权和的强收敛性[J]. 数学学报, 1998, 41(4): 823-832.

*通讯作者。