In the recent literature there has been an increasing interest in regression models for functional variables. Among these there is the situation where the predictor is a random function and the response a scalar.

Let "..." be a linear process with values in a Hilbert space H. We prove a central limit theorem for the vector of empirical covariance operators of the random variables X at orders 0 to h in the space of Hilbert-Schmidt operators.

We consider large and moderate deviations for the empirical mean and covariance of hilbertian autoregressive processes. As an application we obtain moderate deviation principles for the eigenvalues and associated projectors of the empirical covariance sequence.