Rates of convergence and optimal spectral bandwidth for long range dependence

SummaryFor a realization of lengthn from a covariance stationary discrete time process with spectral density which behaves like λ1−2H as λ→0+ for 1/2<H<1 (apart from a slowly varying factor which may be of unknown form), we consider a discrete average of the periodogram across the frequencies 2πj/n,j=1,..., m, wherem→∞ andm/n→0 asn→∞. We study the rate of convergence of an analogue of the mean squared error of smooth spectral density estimates, and deduce an optimal choice ofm.