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

Abstract

Let {V_n, n ≥ 1} be a sequence of random elements in a real separable Banach space and suppose that {V_n, n ≥ 1} is stochastically dominated by a random element V. Let {a_n, n ≥ 1} and {b_n, n ≥ 1} be real sequences with 0 < b_n ↑ ∞. Conditions are provided under which {a_nV_n, n ≥ 1} obeys the general strong law of large numbers almost surely irrespective of the joint distributions of the {V_n, n ≥ 1}. No geometric conditions are imposed on the underlying Banach space. Examples are provided which illustrate, compare, or demonstrate the sharpness of the results.

Keywords:

• Almost sure convergence
• Normed weighted sums
• Real separable Banach space
• Sequence of Banach space valued random elements
• Stochastic dominance
• Strong law of large numbers

Mathematics Subject Classification:

• Primary 60B12
• 60F15
• Secondary 60B11