Copyright © 2009 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

Let {Yi,  1≤i≤n} and {Zi,  1≤i≤n} be sequences of random variables. For any ϵ>0 and a>0, bounds for E(|∑i=1n(Yi+Zi)|−ϵa)+ and E(max⁡1≤k≤n|∑i=1k(Yi+Zi)|−ϵa)+ are obtained. From these results, we establish general methods for obtaining the complete moment convergence. The results of Chow (1988), Zhu (2007), and Wu and Zhu (2009) are generalized and extended from independent (or dependent) random variables to random variables satisfying some mild conditions. Some applications to dependent random variables are discussed.