Let Xn​=∑i=1∞​ai​ϵn−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​,...,xd+1​), d≥0, from
Rd+1 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=1N​(K(Xn​,...,Xn+d​)−EK(Xn​,...,Xn+d​)) are obtained when QN​/N​
obeys the central limit theorem with positive limiting variance.Comment: Published in at http://dx.doi.org/10.3150/07-BEJ112 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm