Statistical inference of spectral estimation for continuous-time MA processes with finite second moments

In this paper we investigate a continuous-time MA (moving average) process (Xt)t≥0 sampled at an equally spaced time grid {∆,2∆, . . . ,n∆}, where the grid distance ∆ > 0 is fixed and n denotes the number of observations, in the frequency domain. We derive for the process (Xk∆)k∈N with finite second moments the asymptotic behavior of the periodogram and of the lag-window spectral density estimator. The periodogram is not a consistent estimator for the spectral density of (Xk∆)k∈N. Different periodogram frequencies are asymptotically independent exponentially distributed like for ARMA processes in discrete time. This result is basic for frequency bootstraps. In contrast, the lag-window spectral density estimator is a consistent estimator for the spectral density of (Xk∆)k∈N and moreover, it is asymptotically normally distributed.