SPECTRAL ESTIMATION OF CONTINUOUS-TIME STATIONARY-PROCESSES FROM RANDOM SAMPLING

Let X = {X (t), - infinity < t < infinity} be a continuous-time stationary process with spectral density function phi(x)(lambda) and {tau(k)} be a stationary point process independent of X. Estimates ($) over cap phi(x)(lambda) of phi(x)(lambda) based on the discrete-time observation {X (tau(k)), tau(k)} are considered. Asymptotic expressions for the bias and covariance of ($) over cap phi(x)(lambda) are derived. A multivariate central limit theorem is established for the spectral estimators ($) over cap phi(x)(lambda). Under mild conditions, it is shown that the bias is independent of the statistics of the sampling point process {tau(k)} and that there exist sampling point processes such that the asymptotic variance is uniformly smaller than that of a Poisson sampling scheme for all spectral densities phi(x)(lambda) and all frequencies lambda.