Rates of convergence and optimal spectral bandwidth for long range dependence (Q1333578)

For a realization of length \(n\) from a covariance stationary discrete time process with spectral density which behaves like \(\lambda^{1-2H}\) as \(\lambda \to 0 +\) for \({1 \over 2} < H < 1\) (apart from a slowly varying factor which may be of unknown form), we consider a discrete average of the periodogram across the frequencies \(2 \pi j/n\), \(j = 1, \dots, m\), where \(m \to \infty\) and \(m/n \to 0\) as \(n \to \infty\). 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 \(m\).