A law of the single logarithm for weighted sums of i.i.d. random elements

Let {X,Xn,n≥1}{X,Xn,n≥1} be a sequence of i.i.d. Banach space valued random elements with E(‖X‖β/(log‖X‖)β/2)<∞E(‖X‖β/(log‖X‖)β/2)<∞, and {ani,1≤i≤n,n≥1}{ani,1≤i≤n,n≥1} an array of constants satisfying ∑i=1n|ani|α=O(n), where α>0α>0, β>0β>0, and 1/2=1/α+1/β1/2=1/α+1/β. In this paper, we obtain a law of the single logarithm for weighted sums ∑i=1naniXi. We also obtain a strong law of large numbers for weighted sums of i.i.d. Banach space valued random elements with a suitable moment condition. No assumptions are made concerning the geometry of the underlying Banach space.