| We study prediction in the functional linear model with functional outputs. We provide the asymptotic mean square prediction error with constants. The rates we obtain are optimal in minimax sense and generalize those found, when the output is real. Conversely to previous works, our main results hold with no prior assumptions on the rate of decay of the eigenvalues of the input. This allows to consider a class of parameters which is wider than those needed in previous papers on this topic. The methods of proofs are based on convex and exponential inequalities for the eigenvalues. 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 and shows that no weak convergence result can be obtained for the bare estimate (without weak topologies or smooth norms). |
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