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