Some Strong Laws for Normed Weighted Sums of Stochastically Dominated Banach Space Valued Random Elements Irrespective of Their Joint Distributions

Let {V-n, n1} be a sequence of random elements in a real separable Banach space and suppose that {V-n, n1} is stochastically dominated by a random element V. Let {a(n), n1} and {b(n), n1} be real sequences with 0<b(n). Conditions are provided under which {a(n)V(n), n1} obeys the general strong law of large numbers almost surely irrespective of the joint distributions of the {V-n, n1}. No geometric conditions are imposed on the underlying Banach space. Examples are provided which illustrate, compare, or demonstrate the sharpness of the results.