On almost certain convergence of double series of random elements and the rate of convergence of tail series

Abstract

ABSTRACT For a double array of random elements taking values in a real separable Banach space, we provide sufficient conditions for the double series to converge almost certainly to a random element S as . For a convergent double series, we study the rate of convergence to a random element S by studying the rate at which the corresponding tail series converges almost certainly to 0 as where .