Paper Title
The Almost Sure Convergence of the Best Linear Predictor for Autoregressive Processes in Hilbert Spaces

Abstract
We consider the Best Linear Predictor (BLP) of Functional Autoregressive Processes built with orthogonal projection on linearly closed subspaces introduced by R.Fortet. We show almost sure convergence and uniform convergence of the predictors. We then provide exponential bounds for the predictors extending. We illustrate the finite sample performance of the predictors by a simulation study and through real examples from climatology comparing with others prediction methods existing in the literature. This simulation study enlightens on the link between the convergence rates of predictors BLP and the presence of the first eigenvalues of the covariance operator and sample size. Index terms - Functional Autoregressive Processes, enclosed subspace, measurable linear transformations, orthogonal projection, El Nino series.