Sieves estimator of the operator of a functional autoregressive process

We consider the estimation of the operator of one-order functional autoregressive process by the sieves method of Grenander in the case of dependent random variables framework. We show the almost sure convergence in Hilbert–Schmidt norm when the operator is of kernel type in Gaussian case afterwards we generalize the results to the Hilbert–Schmidt operator. In the kernel operator type the a.s. convergence is obtained under polynomial growth size improving the logaritmic growth size obtained early. Prediction of continuous time stochastic process is also examined.