Moment Inequalities and Complete Moment Convergence

Let {Y(i), 1 <= i <= n} and {Z(i), 1 <= i <= n} be sequences of random variables. For any epsilon > 0 and a > 0, bounds for E(vertical bar Sigma(n)(i=1) (Y(i) + Z(i))vertical bar - epsilon a)(+) and E(max(1 <= k <= n)vertical bar Sigma(k)(i=1) (Y(i) + Z(i))vertical bar -epsilon a)(+) 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. Copyright (C) 2009 Soo Hak Sung.