Prévision d’un processus autorégressif fonctionnel via les sous espaces clos.

Abstract

We consider the Best Linear Predictor (BLP) of Functional Autoregressive Processes built with orthogonal projection on linearly closed subspaces introduced by R. Fortet (1995). This approach directly focuses on the prediction of this class of processes and we show almost sure convergence and exponential bounds for the predictors BLP. Then we improve the existing results in the literature. We give the almost sure convergence of the predictors BLP for C[0;1]-valued autoregressive process when it ruled by a bounded linear operator. Our conditions essentially carry on the decay rate of the eigenvalues of the covariance operator of the process. We illustrate the finite sample performance of the BLP predictors by a simulation study and through real examples from climatology and consumption of electrical energy. We compare with others prediction methods existing in the literature and enlighten on the link between the convergence rates of BLP predictors and the presence of the first eigenvalues of the covariance operator.