For a sequence of random elements T_n, n ≥ 1 in a real separable Banach space X, we study the notion of T_n converging completely to 0 in mean of order p where p is a positive constant. This notion is stronger than (i) T_n converging completely to 0 and (ii) T_n converging to 0 in mean of order p. When X is of Rademacher type p (1 ≤ p ≤ 2), for a sequence of independent mean 0 random elements V_n, n ≥ 1 in X and a sequence of constants b_n → ∞, conditions are provided under which the normed sum converges completely to 0 in mean of order p. Moreover, these conditions for converging completely to 0 in mean of order p are shown to provide an exact characterization of Rademacher type p Banach spaces. Illustrative examples are provided.