Abstract

Let {Xnk} be an array of rowwise independent random elements in a separable Banach space of type p+δ with EXnk=0 for all k, n. The complete convergence (and hence almost sure convergence) of n1/pk=1nXnk to 0, 1p<2, is obtained when {Xnk} are uniformly bounded by a random variable X with E|X|2p<. When the array {Xnk} consists of i.i.d, random elements, then it is shown that n1/pk=1nXnk converges completely to 0 if and only if EX112p<.