Stochastic convergence of randomly weighted sums of random elements

Abstract

Let {X_n} be random elements in a separable Banach space and let {a_nk} be an array of random variables. Convergence in probability and almost surely is obtained for the weighted sum under varying distributional and moment conditions on the random weights and on the random elements and geometric conditions on Banach spaces. In general, these results include the results for constant weights and real-valued random variables and are motivated in part by estimation problems and consistency considerations. Moreover, similar results are obtained for the space D[0, 1] under varying hypotheses of boundedness conditions on the moments and conditions on the mean oscillation of the random elements {X_n} on subintervals of a partition of [0,1] and represent significant improvements over existing laws o f large numbers and convergence results for weighted sums of random elements in D[0,1].