Convergence of randomly weighted sums of Banach space valued random elements and uniform integrability concerning the random weights

Some notions of uniform integrability of an array of random elements in a separable Banach space with respect to an array of random variables are introduced and characterized, in order to obtain weak laws of large numbers for randomly weighted sums. The paper contains results which generalize some previous results for weighted sums with nonrandom weights, and one of them is used to obtain a result of convergence for sums with a random number of addends. Furthermore, a result of almost everywhere convergence of the sequence of certain conditional expectations of the row sums is obtained.