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Some Strong and Weak Limit Theorems for Weighted Sums of i.i.d. Banach Space Valued Random Elements with Slowly Varying Weights

Abstract For a sequence of independent and identically distributed random elements {V n , n ≥ 1} in a real separable Banach space 𝒳 and a monotone (increasing or decreasing) positive slowly varying function L, it is shown that the normed weighted sum converges almost surely to an element v ∈ 𝒳 if and only if E‖V 1‖ < ∞ and E V 1 = v. No geometric conditions on 𝒳 are required for this result. Moreover, when 𝒳 is of Rademacher type p (1 < p ≤ 2), conditions are provided (which are weaker than E‖V 1‖ < ∞) which ensure that U n converges in probability to some element v ∈ 𝒳. This result can fail without the Rademacher type p hypothesis. Both of these limit theorems are new results even when 𝒳 is the real line.