Limit theorems for weighted sums of random elements in separable Banach spaces

Let a_nk, n ≥ 1, k ≥ 1, be a double array of real numbers and let V_n, n ≥ 1, be a sequence of random elements taking values in a separable Banach space. In this paper, we examine under what conditions the sequence Σ_k≥1a_nkV_k, n ≥ 1, has a limit either in probability or almost surely.