On moments of the supremum of normed weighted averages

Let { X, X n ; n ≥ 1} be a sequence of real‐valued iid random variables, 0 < r < 2 and p > 0. Let D = { A = ( a nk ; 1 ≤ k ≤ n , n ≥ 1); a nk , ϵ R and sup n, k | a n,k | < ∞}. Set S n ( A ) = ∑ n k=1 a n, k X k for A ϵ D and n ≥ 1. This paper is devoted to determining conditions whereby E {sup n ≥ 1 , | S n ( A )|/ n 1/r } p < ∞ or E {sup n ≥ 2 | S n ( A )|/2 n log n ) 1/2 } p < ∞ for every A ϵ 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).