Estimation of the Cross-Spectrum

Assuming that the wide sense stationary process being sampled has an absolutely continuous spectrum, Parzen 5 has shown the consistency of a general class of estimators for estimating the spectral density at a given frequency. He has shown that the class of estimators he considers, contains as a particular case estimators considered earlier by Grenander and Rosenblatt 2, Barlett 1, Tukey 7, 8, Daniell, and Lomnicki and Zaremba 3. In this paper we extend Parzen's results to the case of a Stationary Gaussian Vector process whose spectrum is not necessarily absolutely continuous and show that this general class of estimators consistently estimate the co- and quadrature spectral densities at all those frequencies where they exist. It has been earlier shown by the author 4, using an entirely different approach from the one presented in this paper, that for a normal Stationary process whose spectrum besides the absolutely continuous part contains a step function with a finite number of saltuses, the weighted periodogram estimator, which is a particular case of the general class of estimators considered by Parzen 5, is still a consistent estimate of the spectral density at any point of continuity of the spectrum. Thus this paper also substantially generalizes the earlier result of the author, where he limits himself to a finite number of saltuses.