Mean convergence theorems and weak laws of large numbers for weighted sums of dependent random variables

Under some conditions of uniform integrability and appropriate conditions, mean convergence theorems and weak laws of large numbers for weighted sums of dependent random variables are obtained. Our results extend and improve the results of [H.S. Sung, S. Lisawadi, A. Volodin, Weak laws of large numbers for arrays under a condition of uniform integrability, J. Korean Math. Soc. 45 (2008) 289–300] and [M. Ordóñez Cabrera, A. Volodin, Mean convergence theorems and weak laws of large numbers for weighted sums of random variables under a condition of weighted integrability, J. Math. Anal. Appl. 305 (2005) 644–658].