Convergence in the pth-mean and some Weak Laws of Large Numbers for weighted sums of random elements in separable normed linear spaces

. Let Xn, n >= 1, be a sequence of tight random elements taking values in a separable Banach space B such that Xn, n >= 1, is uniformly integrable. Let ank, n >= 1, k >= 1, be a double array of real numbers satisfying [Sigma]k >= 1 ank = 1 for some positive constant [Gamma]. Then [Sigma]k >= 1 ankXk, n >= 1, converges to 0 in probability if and only if [Sigma]k >= 1 ankf(Xk), n >= 1, converges to 0 in probability for every f in the dual space B*.