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Estimation and prediction for infinite dimensional autoregressive processes

In recent years much effort are devoted in order to predict the evolution of a discrete time stochastic processes. Few is known on the prediction of a continuous time processes on an entire time interval. For this task we introduce the p-order Banach valued autoregressive processes which appear to be a natural extension of the well-known R d -valued autoregressive processes to a functional framework. We estimate consistently the parameters of a Hilbertian autoregressive processes when are unknown and we specify the rates. In a wide range of applications this model can be used and is able to handle the prediction sucessfully.