On almost sure and mean convergence of normed double sums of banach space valued random elements

For a double array of independent mean 0 random elements {V-mn, m >= 1 >= n >= 1} in a real separable Rademacher type p (1 <= p <= 2) Banach space and constants alpha > 0 and beta > 0, the main result shows that if the double series Sigma(infinity)(m=1) Sigma(infinity)(n=1) E parallel to V-mn parallel to(p)/m(alpha p)pn(beta p) converges, then max(1 <= k <= m), 1 <= l <= n parallel to Sigma(k)(i=1) Sigma(l)(j=1) V-ij parallel to/m(alpha)n(beta) converges to 0 almost surely and in mean of order p as max {m, n} -> infinity. This implication provides a characterization of Rademacher type p Banach spaces. Related results are also obtained for (i) normed double sums from a double array of random elements without imposing any geometric condition on the Banach space or any independence or mean 0 conditions on the random elements, and (ii) normed double sums from a p-orthogonal double array of random elements in a Rademacher type p (1 <= p <= 2) Banach space. The sharpness of the main result is illustrated by an example.