A new type of compact uniform integrability with application to degenerate mean convergence of weighted sums of Banach space valued random elements

In this correspondence, for an array {Xnk : un < k < E N} of integrable random elements in a real separable Banach space and an array {a,, : u < k < n E N} of real numbers, a new type of compact uniform integrability is introduced and it is used to obtain degenerate mean convergence theorems for the weighted V, sums E a k(X.,,,k n E N. More specifically, conditions are provided k=u under which lim E n-400 E a,,k(Xnk EX k) k=u, = 0. 2020 Elsevier Inc. All rights reserved.