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

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

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