Some mean convergence theorems are established for randomly weighted sums of the form ∑_j = 1^k_n A_njV_nj and ∑_j = 1^T_n A_njV_nj where A_nj, j ≥ 1, n ≥ 1 is an array of random variables, V_nj, j ≥ 1, n ≥ 1 is an array of mean 0 random elements in a separable real Rademacher type p (1 ≤ p ≤ 2) Banach space, and k_n, n ≥ 1 and T_n, n ≥ 1 are sequences of positive integers and positive integer-valued random variables, respectively. The results take the form   or   where 1 ≤ r ≤ p. It is assumed that the array A_njV_nj, j ≥ 1, n ≥ 1 is comprised of rowwise independent random elements and that for all n ≥ 1, A_nj and V_nj are independent for all j ≥ 1 and T_n and A_njV_nj, j ≥ 1 are independent. No conditions are imposed on the joint distributions of the random indices T_n, n ≥ 1. The sharpness of the results is illustrated by examples.