On almost sure convergence of weighted sums of random element sequences

We mainly study the almost sure limiting behavior of weighted sums of the form where {Xn, n ≥ 1} is an arbitrary Banach space valued random element sequence or Banach space valued martingale difference sequence and {an, n ≥ 1} and {bn, n ≥ 1} are two sequences of positive constants. Some new strong laws of large numbers for such weighted sums are proved under mild conditions.