On Berry–Esseen bounds for (original) (raw)

∞ 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 −β) 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,. .. , x d+1), d ≥ 0, from R d+1 to R, which includes indicator functions and polynomials, the stationary sequence K(Xn, Xn+1,. .. , X n+d) is considered. By developing a finite orthogonal expansion of K(Xn,. .. , X n+d), the Berry-Esseen type bounds for the normalized sum QN / √ N , QN = N n=1 (K(Xn,. .. , X n+d) − EK(Xn,. .. , X n+d)) are obtained when QN / √ N obeys the central limit theorem with positive limiting variance.