  In this paper, we consider the Whittle estimator for the parameters of a
stationary solution of a continuous-time linear state space model sampled at
low frequencies. In our context the driving process is a L\'evy process which
allows flexible margins of the underlying model. The L\'evy process is supposed
to have finite second moments. It is well known that then the class of
stationary solutions of linear state space models and the class of multivariate
CARMA processes coincides. We prove that the Whittle estimator, which is based
on the periodogram, is strongly consistent and asymptotically normally
distributed. A comparison with the classical setting of discrete-time ARMA
models shows that in the continuous-time setting the limit covariance matrix of
the Whittle estimator has an additional correction term for non-Gaussian
models. For the proof, we investigate as well the asymptotic normality of the
integrated periodogram which is interesting for its own. It can be used to
construct goodness of fit tests. Furthermore, for univariate state space
processes, which are CARMA processes, we introduce an adjusted version of the
Whittle estimator and derive as well the asymptotic properties of this
estimator. The practical applicability of our estimators is demonstrated
through a simulation study.
