Abstract
We study prediction in the functional linear model with functional outputs, Y = SX + ε, where the covari-ates X and Y belong to some functional space and S is a linear operator. We provide the asymptotic mean square prediction error for a random input with exact constants for our estimator which is based on the functional PCA of X. As a consequence we derive the optimal choice of the dimension k n of the projection space. The rates we obtain are optimal in minimax sense and generalize those found when the output is real. Our main results hold for class of inputs X(·) that may be either very irregular or very smooth. We also prove a central limit theorem for the predictor. We show that, due to the underlying inverse problem, the bare estimate cannot converge in distribution for the norm of the function space.