Spectral estimation of continuous-time stationary processes from random sampling

AbstractLet X = {X(t), − ∞ < t < ∞} be a continuous-time stationary process with spectral density function φX(λ) and {τk} be a stationary point process independent of X. Estimates φ̂X(λ) of φX(λ) based on the discrete-time observation {X(τk), τk} are considered. Asymptotic expressions for the bias and covariance of φ̂X(λ) are derived. A multivariate central limit theorem is established for the spectral estimators φ̂X(λ). Under mild conditions, it is shown that the bias is independent of the statistics of the sampling point process {τk} 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 φX(λ) and all frequencies λ