Discrete-Time Spectral Estimation of Continuous-Time Processes-The Orthogonal Series Method

Abstract

Let $\{X(t), - \infty < t < \infty\}$ be a stationary time series with spectral density function $\phi(\lambda).$ Let $\{t_n\}$ be a stationary Poisson point process on the real line. The existence of consistent estimates of $\phi(\lambda)$ based on the discrete-time observations $\{X(t_n)\}^N_{n = 1},$ when the actual sampling times are not known, has been an open question (Beutler). Using an orthogonal series method, a class of spectral estimates is considered and its uniform and integrated uniform consistency in quadratic mean is established. Rates of convergence are established and are compared with the optimal rates of the available (Brillinger, Masry) kernel-type estimates based on the observations $\{X(t_n), t_n\}^N_{n = 1}$.