On the Weak Limiting Behavior of Almost Surely Convergent Row Sums from Infinite Arrays of Rowwise Independent Random Elements in Banach Spaces

For an array {V_nk,k1,n1} of rowwise independent random elements in a real separable Banach space with almost surely convergent row sums , we provide criteria for S_n–A_n to be stochastically bounded or for the weak law of large numbers to hold where {A_n,n1} is a (nonrandom) sequence in .