Mean convergence theorems and weak laws of large numbers for weighted sums of random variables under a condition of weighted integrability

From the classical notion of uniform integrability of a sequence of random variables, a new concept of integrability (called h-integrability) is introduced for an array of random variables, concerning an array of constants. We prove that this concept is weaker than other previous related notions of integrability, such as Cesaro uniform integrability [Chandra, Sankhya Ser. A 51 (1989) 309-317], uniform integrability concerning the weights [Ordonez Cabrera, Collect. Math. 45 (1994) 121-132] and Cesaro alpha-integrability [Chandra and Goswami, J. Theoret. Probab. 16 (2003) 655-669]. Under this condition of integrability and appropriate conditions on the array of weights, mean convergence theorems and weak laws of large numbers for weighted sums of an array of random variables are obtained when the random variables are subject to some special kinds of dependence: (a) rowwise pairwise negative dependence, (b) rowwise pairwise non-positive correlation, (c) when the sequence of random variables in every row is phi-mixing. Finally, we consider the general weak law of large numbers in the sense of Gut [Statist. Probab. Lett. 14 (1992) 49-52] under this new condition of integrability for a Banach space setting. (c) 2004 Elsevier Inc. All rights reserved.