Precise Rates in Complete Moment Convergence for Negatively Associated Sequences

Title & Authors
Precise Rates in Complete Moment Convergence for Negatively Associated Sequences
Ryu, Dae-Hee;

Abstract
Let {$\small{X_n}$, n $\small{{\ge}}$ 1} be a negatively associated sequence of identically distributed random variables with mean zeros and positive finite variances. Set $\small{S_n}$ = $\small{{\Sigma}^n_{i=1}\;X_i}$. Suppose that 0 < $\small{{\sigma}^2=EX^2_1+2{\Sigma}^{\infty}_{i=2}\;Cov(X_1,\;X_i)}$ < $\small{{\infty}}$. We prove that, if $\small{EX^2_1(log^+{\mid}X_1{\mid})^{\delta}}$ < $\small{{\infty}}$ for any 0< $\small{{\delta}{\le}1}$, then $\small{\lim_{{\epsilon}\downarrow0}{\epsilon}^{2{\delta}}\sum_{{n=2}}^{\infty}\frac{(logn)^{\delta-1}}{n^2}ES^2_nI({\mid}S_n{\mid}\geq{\epsilon}{\sigma}\sqrt{nlogn}=\frac{E{\mid}N{\mid}^{2\delta+2}}{\delta}}$, where N is the standard normal random variable. We also prove that if $\small{S_n}$ is replaced by $\small{M_n=max_{1{\le}k{\le}n}{\mid}S_k{\mid}}$ then the precise rate still holds. Some results in Fu and Zhang (2007) are improved to the complete moment case.
Keywords
Precise rates;complete moment convergence;negatively associated;law of the logarithm;
Language
English
Cited by
References
1.
Alam, K. and Saxena, K. M. L. (1981). Positive dependence in multivariate distributions, Communi-cations in Statistics - Theory and Methods, 10, 1183-1196

2.
Joag-Dev, K. and Proschan, F. (1983). Negative association of random variables with applications, The Annals of Statisicis, 11, 286-295

3.
Fu, K. A. and Zhang, L. X. (2007). Precise rates in the law of the logarithm for negatively associated random variables, Computers & Mathematics with Applications, 54, 687-698

4.
Huang, W. and Zhang, L. X. (2005). Precise rates in the law of the logarithm in the Hilbert space, Journal of Mathematical Analysis and Applications, 304, 734-758

5.
Kim, T. S., Lee, S. W. and Ko, M. H. (2001). On the estimation of empirical distribution function for negatively associated processes, The Korean Communication in Statistics, 8, 229-236

6.
Ko, M. H. (2009). A central limit theorem for the linear process in a Hilbert space under negative association, Communication of the Korean Statistical Society, 16, 687-696

7.
Li, Y. X. and Zhang, L. X. (2004). Complete moment convergence of moving- average processes under dependence assumptions, Statistics & Probability Letters, 70, 191-197

8.
Liang, H. Y. (2000). Complete convergence for weighted sums of negatively associated random vari-ables, Statistics & Probability Letters, 48, 317-325

9.
Liu, W. D. and Lin, Z. Y. (2006). Precise asymptotics for a new kind of complete moment conver-gence, Statistics & Probability Letters, 76, 1787-1799

10.
Newman, C. M. (1984). Asymptotic independence and limit theorems for positively and negatively dependent random variables, Y. L. Tong, ed. Inequalities in Statistics and Probability: Proceed-ings of the Symposium on Inequalities in Statistics and Probability, 5, 127-140

11.
Peligrad, M. (1987). The convergence of moments in the central limit theorem for \rho-mixing sequence of random variables, In Proceedings of the American Mathematical Society, 101, 142-148

12.
Shao, Q. M. and Su, C. (1999). The law of the iterated logarithm for negatively associated random variables, Stochastic Processes and their Applications, 83, 139-148

13.
Shao, Q. M. (2000). A comparison theorem on moment inequalities between negatively associated and independent random variables, Journal of Theoretical Probability, 13, 343-356

14.
Zhao, Y. (2008). Precise rates in complete moment convergence for \rho-mixing sequences, Journal of Mathematical Analysis and Applications, 339, 553-565