Empirical spectral processes for stationary state space models

In this paper, we consider function-indexed normalized weighted integrated periodograms for equidistantly sampled multivariate continuous-time state space models which are ARMA processes. Thereby, the sampling distance is fixed and driving Lévy process has at least a finite fourth moment. Under different assumptions on function moments of derive central limit theorem periodogram. Either assumption or existence weaker. Furthermore, show weak convergence in both continuous functions dual to Gaussian give an explicit representation covariance function. The results can be used asymptotic behavior Whittle estimator construct goodness-of-fit test statistics as Grenander–Rosenblatt statistic Cramér–von Mises statistic. We present exact distributions their performance through simulation study.