Convergence of randomly weighted sums for arrays under a condition of integrability

Let {X
ni
, u
n
≤ i ≤ ν
n
, n ≥ 1} and {A
ni
, u
n
≤ i ≤ ν
n
, n ≥ 1} be two arrays of random variables. A new notion of integrability for an array of random variables {X
ni
} with respect to an array of random variables {A
ni
} is introduced, weak laws of large numbers for the randomly weighted sums equation are obtained. Our results extend and sharpen the known results in the literature.