Limit theorems for randomly weighted sums of random elements in normed linear spaces

Let {Vn; n >= 1} be a sequence of random elements in a separable normed linear space E, uniformly dominated by a random variable V. Let {Ank; K = 1, 2, ..., n; n >= 1} be a triangular array of random variables. In this paper, conditions for the convergence of [Sigma]k=1n AnkVk to zero (in probability and completely) are obtained. No geometric condition on E is imposed.