Weak convergence for the covariance operators of a Hilbertian linear process

Let Xt=[summation operator]k=-[infinity]+[infinity]ak([var epsilon]t-k) be a linear process with values in a Hilbert space H. The H valued r.v. [var epsilon]k are i.i.d. centered, the ak's 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 in the space of Hilbert-Schmidt operators. Statistical applications are given in the area of principal component analysis for vector dependent random curves.