A Consistent Estimate of the Spectrum by Random Sampling of the Time Series

A new estimation scheme of the spectral density function of a stationary Gaussian time series using observations at discrete times is presented. For a broad class of spectral densities, asymptotic expressions for the bias and variance of the estimate are derived. It is shown that the estimate is consistent even though the time series is not assumed to be band-limited. It is further shown that the consistency of the estimate holds for all positive values of the average sampling rate.