Weak convergence for the covariance operators of a Hilbertian linear process

Let Xt=∑k=−∞+∞ak(εt−k) be a linear process with values in a Hilbert space H. The H valued r.v. εk are i.i.d. centered, the akʹs are linear operators. We prove a central limit theorem for the vector of empirical covariance operators of the random variables Xt at orders 0 to h∈N in the space of Hilbert–Schmidt operators. Statistical applications are given in the area of principal component analysis for vector dependent random curves.