Asymptotics of prediction in functional linear regression with functional outputs

We study prediction in the functional linear model with functional outputs : $Y=SX+\epsilon $ where the covariates $X$ and $Y$ belong to some functional space and $S$ is a linear operator. We provide the asymptotic mean square prediction error with exact constants for our estimator which is based on functional PCA of the input and has a classical form. 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 with no prior assumptions on the rate of decay of the eigenvalues of the input. This allows to consider a wide class of parameters and inputs $X(\cdot) $ that may be either very irregular or very smooth. We also prove a central limit theorem for the predictor which improves results by Cardot, Mas and Sarda (2007) in the simpler model with scalar outputs. We show that, due to the underlying inverse problem, the bare estimate cannot converge in distribution for the norm of the function space