On Berry–Esseen bounds for non-instantaneous filters of linear processes

∞ i=1 aiεn−i, where the εi are i.i.d. with mean 0 and at least finite second moment, and the ai are assumed to satisfy |ai| = O(i) with β > 1/2. When 1/2 < β < 1, Xn is usually called a long-range dependent or long-memory process. For a certain class of Borel functions K(x1, . . . , xd+1), d≥ 0, from R to R, which includes indicator functions and polynomials, the stationary sequence K(Xn,Xn+1, . . . ,Xn+d) is considered. By developing a finite orthogonal expansion of K(Xn, . . . ,Xn+d), the Berry–Esseen type bounds for the normalized sum QN/ √ N,QN = ∑N n=1 (K(Xn, . . . ,Xn+d)−EK(Xn, . . . ,Xn+d)) are obtained when QN/ √ N obeys the central limit theorem with positive limiting variance.