Almost sure convergence of weighted sums of random elements in Banach spaces

Abstract

Let {Vk∶k≥1} be a sequence of random elements in a real separable normed linear space X, and let {ank∶n≥1, k≥1} be an array of real numbers. Several theorems are given which provide conditions for the convergence with probability one of \(s_n = \sum\nolimits_{k = 1}^n {a_{nk} V_k } \)to the zero element of X. One result states that if X is B-convex and if the random elements are independent with expected values zero and uniformly bounded rth moments for some r>1, then, under a given set of conditions on {ank}, Sn→0 in X with probability one.KeywordsBanach SpaceIndependent Random VariableRandom ElementZero ElementNormed Linear SpaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.