Titelaufnahme

We study the distributional properties of a quadratic form of a stationary functional time series under mild moment conditions. As an important application, we obtain consistency rates of estimators of spectral density operators andprove joint weak convergence to a vector of complex Gaussian random operators. Weak convergence is established based on an approximation of the form via transforms of Hilbert-valued martingale difference sequences. As a side-result, the distributional properties of the long-run covariance operator are established.