Journal of Inequalities and Applications
Volume 2008 (2008), Article ID 598319, 10 pages
doi:10.1155/2008/598319
Research Article
Maximal Inequalities for Dependent Random Variables and Applications
Department of Applied Mathematics, Pai Chai University, Taejon 302-735, South Korea
Received 16 April 2008; Revised 3 June 2008; Accepted 7 July 2008
Academic Editor: Ondrej Dosly
Copyright © 2008 Soo Hak Sung. 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
For a sequence {Xn,n≥1} of dependent square integrable random variables and
a sequence {bn,n≥1} of positive numbers, we establish a maximal inequality for
weighted sums of dependent random variables. Applying this inequality, we obtain the almost sure
convergence of ∑i=1nXi/bi and ∑i=1nXi/bn.