Summary: We study problems of semiparametric statistical inference connected with long-memory covariance stationary time series, having spectrum which varies regularly at the origin: There is an unknown self-similarity parameter, but elsewhere the spectrum satisfies no parametric or smoothness conditions, it need not be in \(L_ p\), for any \(p>1\), and in some circumstances the slowly varying factor can be of unknown form. The basic statistic of interest is the discretely averaged periodogram, based on a degenerating band of frequencies around the origin.
We establish some consistency properties under mild conditions. These are applied to show consistency of new estimates of the self-similarity parameter and scale factor. We also indicate applications of our results to standard errors of least squares estimates of polynomial regression with long-memory errors, to generalized least squares estimates of this model and to estimates of a “cointegrating” relationship between long- memory time series.