Convergence in mean of weighted sums of {an,k}-compactly uniformly integrable random elements in Banach spaces

The convergence in mean of a weighted sum ∑kank(Xk−EXk) of random elements in a separable Banach space is studied under a new hypothesis which relates the random elements with their respective weights in the sum: the {ank}-compactly uniform integrability of {Xn}. This condition, which is implied by the tightness of {Xn} and the {ank}-uniform integrability of {‖Xn‖}, is weaker than the compactly miform integrability of {Xn} and leads to a result of convergence in mean which is strictly stronger than a recent result of Wang, Rao and Deli