ON ALMOST SURE CONVERGENCE FOR WEIGHTED SUMS OF LNQD RANDOM VARIABLES

• Journal title : Honam Mathematical Journal
• Volume 34, Issue 2,  2012, pp.241-252
• Publisher : The Honam Mathematical Society
• DOI : 10.5831/HMJ.2012.34.2.241
Title & Authors
ON ALMOST SURE CONVERGENCE FOR WEIGHTED SUMS OF LNQD RANDOM VARIABLES
Choi, Jeong-Yeol; Kim, So-Youn; Baek, Jong-Il;

Abstract
Let $\small{\{X_{ni},\;1{\leq}i{\leq}n,\;n{\geq}1\}}$ be a sequence of LNQD which are dominated randomly by another random variable X. We obtain the complete convergence and almost sure convergence of weighted sums $\small{{\sum}^n_{i=1}a_{ni}X_{ni}}$ for LNQD by using a new exponential inequality, where $\small{\{a_{ni},\;1{\leq}i{\leq}n,\;n{\geq}1\}}$ is an array of constants. As corollary, the results of some authors are extended from i.i.d. case to not necessarily identically LNQD case.
Keywords
Strong law of large numbers;almost sure convergence;arrays;linearly negative quadrant random variables;
Language
English
Cited by
1.
Complete convergence for weighted sums of pairwise independent random variables, Open Mathematics, 2017, 15, 1
References
1.
Alam, K., and Saxena,K. M. L.(1981). Positive dependence in multivariate distributions. Commun. Statist. Theor. Meth, A10, 1183-1196.

2.
Baek, J. I., Niu, S.L., Lim, P. K., Ahn, Y. Y. and Chung, S.M.(2005). Almost sure convergence for weighted sums of NA random variables. Journal of the Korean Statistical Society, 32(4), 263-272.

3.
Baek, J. I., Park, S.T., Chung, S. M., Liang, H. Y. and Lee, C. Y.(2005). On the complete convergence of weighted sums for dependent random variables. Journal of the Korean Statistical Society, 34(1), 21-33.

4.
Baek, J. I., Park, S.T., Chung, S. M., and Seo, H. Y.(2005). On the almost sure convergence of weighted sums of negatively associated random variables. Comm.Kor.Math.Soci 20(3), 539-546.

5.
Bai, Z. D. and Cheng, P. E.(2000). Marcinkiewicz strong laws for linear statistics. Statist. & Probab. Lett. 46, 105-112.

6.
Cai, Z. and Roussas, G. G.(1997). Smooth estimate of quantiles under association. Stat.& Probab.Lett. 36, 275-287.

7.
Chow, Y. S. and Lai, T. L.(1973). Limit behaviour of weighted sums of independent random variables. Ann. Probab. 1, 810-824.

8.
Cuzick, J.(1995). A strong law for weighted sums of i.i.d. random variables. J. Theoret. Probab. 8, 625-641.

9.
Joag-Dev, K. and Proschan, F.(1983). Negative association of random variables with applications. Ann. Statist. 11, 286-295.

10.
Karlin,S., Rinott,R.(1980). Classes of ordering measures and related correlation inequalities. II. Multivariate reverse rule distributions. Journal of Multivariate Analy. 10 , 499-516.

11.
Ko, M. H., Ryu, D. H., & Kim, T. S.(2007). Limiting behaviors of weighted sums for linearly negative quadrant dependent random variables. Taiwanese Journal of Mathematics, 11(2), 511-522.

12.
Liang, H. Y., Zhang, D. X., and Baek, J.L.(2004). Convergence of weighted sums for dependent random variables. Jour.Kor.Math.Soc., 41(5), 883-894.

13.
Liang,H. Y.(2000). Complete convergence for weighted sums of negatively associated random variables. Statist.&Probab.Lett. 48 , 317-325.

14.
Lehmann, E. L.(1966). Some concepts of dependence. The Annals of Mathematical Statistics, 37, 1137-1153.

15.
Matula,P.(1992). A note on the almost sure convergence of sums of negatively dependent random variables. Statist. Probab. Lett. 15 , 209-213.

16.
Newman, C. M.(1984). Asymptotic independence and limit theorems for positively and negatively dependent random variables. In Y. L. Tong(Ed.)., Statistics and probability, vol. 5(pp. 127-140). Hayward, CA: Inst. Math. Statist.

17.
Roussas, G. G.(1994). Asymptotic normality of random fields of positively or negatively associated processes. J. Multiv. Analysis, 50 , 152-173.

18.
Shao,Q. M., Su, C.(1999). The law of the iterated logarithm for negatively associated random variables. Stochastic Process Appl. 83 , 139-148.

19.
Su,C. Qin,Y. S.(1997). Limit theorems for negatively associated sequences. Chinese Science Bulletin, 42, 243-246.

20.
Sung, S.(2001). Strong laws for weighted sums of i.i.d.random variables. Statist. & Probab. Lett. 52, 413-419.

21.
Wang, J., & Zhang, L.(2006). A Berry-Esseen theorem for weakly negatively dependent random variables and its applications. Acta Mathematica Hungarica, 110(4), 293-308.