On convergence of series of independent random elements in banach spaces

Abstract

The rate of convergence for an almost certainly convergent series independent random elements in a real separable Banach space is studied in this paper. More specifically, when S n converges almost certainly to a random element S, the tail series is a well defined sequence of random elements with almost certainly. The main result establishes for a sequence of positive constants with b j ⩽ Const. b n whenever the equivalence between the tail series weak law of large numbers and the limit law thereby extending a result of Nam and Rosalsky [20] to a Banach space setting while also simplifying the argument used in the earlier result. The quasimonotonicity proviso on cannot be dispensed with