Matrix Transformations of Weakly Dependent Random Variables

Let X 0 , X 1 … be random variables satisfying the inequalityE|∑ k = 0 nc kX k | r ⩽ C(∑ k = 0 n| c k | p)r / pfor every sequence c 0 , c 1 ,…, of coefficients and for every n = 0, 1,…, where 1 < p ⩽ 2, r > p and C > 0 are constants. By the aid of a summability matrix T = {a nk :n, k = 0, 1,…} we form the meansT n = ∑ k = 0 ∞a n kX kfor n = 0, 1, …. We prove that T n → 0 almost surely as n → ∞ under fairly general conditions on {a nk }. Our result contains as special cases an earlier result of D. Borwein and another result stating a strong law of large numbers for the random variables X k .