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.