Poisson sampling and spectral estimation of continuous-time processes

A class of spectral estimates of continuous-time stationary stochastic processes from a finite number of observations taken at Poisson sampling instants is considered. The asymptotic bias and covariance of the estimates are derived, and the influence of the spectral windows and the sampling rate on the performance of the estimates is discussed. The estimates are shown to be consistent under mild smoothness conditions on the spectral density. Comparison is made with a related class of spectral estimates suggested in [15] where the number of observations is {em random}. It is shown that the periodograms of the two classes have distinct statistics.