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

Let X_n = Σ_i∞=1 a_iε _n-_i, where the ε_i are i.i.d, with mean 0 and at least finite second moment, and the a_i are assumed to satisfy a_i = O(_i^-β) with β > 1/2. When 1/2 < β < 1, X_n is usually called a long-range dependent or long-memory process. For a certain class of Borel functions K(x₁,..., x_d+1), d ≥0, from R^d+1 to R, which includes indicator functions and polynomials, the stationary sequence K (X_n, X_n+1,..., X_n+d) is considered. By developing a finite orthogonal expansion of K (X_n,..., X_n+d), the Berry-Esseen type bounds for the normalized sum Q_N/√N, Q_N = ε _n^N=₁ (K(X_n,..., X_n+d) - EK(X_n,..., X_n+d)) are obtained when Q_N/√N obeys the central limit theorem with positive limiting variance.