Journal of Inequalities and Applications
Volume 2008 (2008), Article ID 385362, 11 pages
doi:10.1155/2008/385362
Research Article

Exponential Inequalities for Positively Associated Random Variables and Applications

Guodong Xing,1 Shanchao Yang,2 and Ailin Liu3

1Department of Mathematics, Hunan University of Science and Engineering, Yongzhou, 425100 Hunan, China
2Department of Mathematics, Guangxi Normal University, Guilin, 541004 Guangxi, China
3Department of Physics, Hunan University of Science and Engineering, Yongzhou, 425100 Hunan, China

Received 1 January 2008; Accepted 6 March 2008

Academic Editor: Jewgeni Dshalalow

Copyright © 2008 Guodong Xing et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We establish some exponential inequalities for positively associated random variables without the boundedness assumption. These inequalities improve the corresponding results obtained by Oliveira (2005). By one of the inequalities, we obtain the convergence rate n−1/2(log⁡log⁡n)1/2(log⁡n)2 for the case of geometrically decreasing covariances, which closes to the optimal achievable convergence rate for independent random variables under the Hartman-Wintner law of the iterated logarithm and improves the convergence rate n−1/3(log⁡n)5/3 derived by Oliveira (2005) for the above case.