Given a triangular array X n,k , 1 k n, n 1 of random variables satisfying E|X n,k | p < ∞ for some p 1 and sequences {b n }, {c n } of positive real numbers, we shall prove that ∞ n=1 c n E n k=1 (X n,k − E X n,k) /b n − ε p + < ∞, where x + = max(x, 0). Our results are announced in a general setting, allowing us to obtain the convergence of the series in question under various types of dependence.