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Let (Xk, k = ..., -1, 0, 1, . . . ) be a weakly stationary sequence of random variables on a probability space (Q, C, P). That is, EX,2 < oo V k, there are constants IL and y(O) such that EXk = IL and Var Xk = y(O) V k, and there is a sequence of numbers -y(l), y(2), y(3), . . . such that Cov(Xk, XI) = y(lk 11) V k # 1. For -oo < J S L S oo let HfL denote the LV-closure of the linear space spanned by the r.v.'s Xk-li, J S k < L. For n = 1, 2, 3, . . . let Sn = XI + X2 + * + Xn and define the quantities

A sufficient condition for linear growth of variances in a stationary random sequence