Strong laws of large numbers and mean convergence theorems for randomly weighted sums of arrays under a condition of integrability

Let {Xni,un⩽i⩽vn,n⩾1}{Xni,un⩽i⩽vn,n⩾1} and {Ani,un⩽i⩽vn,n⩾1}{Ani,un⩽i⩽vn,n⩾1} be two arrays of random variables. A new concept of integrability for an array of random variables {Xni}{Xni} with respect to an array of random variables {Ani}{Ani} is introduced. Under these notions of integrability and appropriate conditions on the array of random weights, strong laws of large numbers and mean convergence theorems for the randomly weighted sums ∑i=unvn(AniXni−EAniXni) are obtained. Our results extend and sharpen the known results in the literature.