Convergence of weighted sums of tight random elements

Convergence of weighted sums of tight random elements {Vn} (in a separable Banach space) which have zero expected values and uniformly bounded rth moments (r > 1) is obtained. In particular, if {ank} is a Toeplitz sequence of real numbers, then [Sigma]k=1[infinity] ankf(Vk) --> 0 in probability for each continuous linear functional f if and only if ||[Sigma]k=1[infinity] ankVk ||--> 0 in probability. When the random elements are independent and max1<=k<=n ank = (n-8) for some 0 < 1/s < r - 1, then [Sigma]k=1[infinity] ankVk ||--> 0 with probability 1. These results yield laws of large numbers without assuming geometric conditions on the Banach space. Finally, these results can be extended to random elements in certain Fréchet spaces.