Some Banach Autoregressive Estimation Problems

Abstract

In this thesis, we have focused on functional autoregressive process with seasonality. We have provide an estimator of a seasonality perturbed by a continuous time process admitting a C[0,1]-valued autoregressive representation and established its almost sure convergence, asymptotic normality and compact iterated logarithm law. Hence using the compact iterated logarithm law, we have construct a confidence balls for the seasonality in the space C[0,1]. Then, we have studied an estimator of seasonality when it belongs to a finite dimensional space and gave its asymptotic properties. The dimension of this space was therefore estimated when has been considered as being unknown. We concluded by giving numerical simulations that illustrate asymptotic results of the estimators.