RATES OF CONVERGENCE AND OPTIMAL SPECTRAL BANDWIDTH FOR LONG-RANGE DEPENDENCE

For a realization of length n from a covariance stationary discrete time process with spectral density which behaves like lambda1 - 2H as lambda --> 0 + for 1/2 < H < 1 (apart from a slowly varying factor which may bc of unknown form), we consider a discrete average of the periodogram across the frequencies 2 pij/n, j = 1, ..., m, where m --> infinity and m/n --> 0 as n --> infinity. We study the rate of convergence of an analogue of the mean squared error of smooth spectral density estimates, and deduce an optimal choice of m.