Abstract: | Let {X, Xn; n ≥ 1} be a sequence of real-valued iid random variables, 0 < r < 2 and p > 0. Let D = { A = (ank; 1 ≤ k ≤ n, n ≥ 1); ank, ? R and supn, k |an,k| < ∞}. Set Sn( A ) = ∑nk=1an, kXk for A ? D and n ≥ 1. This paper is devoted to determining conditions whereby E{supn ≥ 1, |Sn( A )|/n1/r}p < ∞ or E{supn ≥ 2 |Sn( A )|/2n log n)1/2}p < ∞ for every A ? D. This generalizes some earlier results, including those of Burkholder (1962), Choi and Sung (1987), Davis (1971), Gut (1979), Klass (1974), Siegmund (1969) and Teicher (1971). |