Complete convergence for weighted sums of arrays of random elements

Let { X n k : k , n = 1 , 2 , … } be an array of row-wise independent random elements in a separable Banach space. Let { a n k : k , n = 1 , 2 , … } be an array of real numbers such that ∑ k = 1 ∞ | a n k | ≤ 1 and ∑ n = 1 ∞ exp ( − α / A n ) < ∞ for each α ϵ R + where A n = ∑ k = 1 ∞ a n k 2 . The complete convergence of ∑ k = 1 ∞ a n k X n k is obtained under varying moment and distribution conditions on the random elements. In particular, laws of large numbers follow for triangular arrays of random elements, and consistency of the kernel density estimates is obtained from these results.