Convergence of weighted sums of random variables and uniform integrability concerning the weights

Abstract

Let {ank}, n, k ∈ N be an array of real constants, and let {Xn} be a sequence of random variables. The concept of {ank}-uniform integrability of {Xn} is defined and two characterizations of this concept are established. Limit theorems for weighted sums Σk ank(Xk − EXk) are obtained, when the sequence {Xn} is {ank}-uniformly integrable.