We study 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 and prove 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.

翻译：我们研究在温和时刻条件下固定功能时间序列的二次形式分配特性。作为一项重要应用,我们获得了光谱密度操作员估计值的一致率,并证明与复合高西亚随机操作员矢量的合并程度是弱的。弱的趋同建立在通过Hilbert价值的马丁格尔差异序列变换而使形式接近的基础上。作为副作用,确定了长期共变操作员的分布特性。