Stochastic convergence of weighted sums in normed linear spaces

Let X be a (real) separable Banach space, let {Vk} be a sequence of random elements in X, and let {ank} be a double array of real numbers such that limn-->[infinity] ank = 0 for all k and [Sigma][infinity]k=1 ank <= 1 for all n. Define Sn = [Sigma]nk=1 ank(Vk - EVk). The convergence of {Sn} to zero in probability is proved under conditions on the coordinates of a Schauder basis or on the dual space of X and conditions on the distributions of {Vk}. Convergence with probability one for {Sn} is proved for separable normed linear spaces which satisfy Beck's convexity condition with additional restrictions on {ank} but without distribution conditions for the random elements {Vk}. Finally, examples of arrays {ank}, spaces, and applications of these results are considered.