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 (X _t ) _t≥0 sampled at an equally spaced time grid {Δ,2Δ, …, nΔ}, where the grid distance Δ > 0 is fixed and n denotes the number of observations, in the frequency domain. We derive for the process (X _kΔ) _k∈? 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 (X _kΔ) _k∈?. Different periodogram frequencies are asymptotically independent and 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 (X _kΔ) _k∈? and moreover, it is asymptotically normally distributed.