Optimal rate for covariance operator estimators of functional autoregressive processes with random coefficients

We improve a result of Allam and Mourid (2014) by deriving the optimal n rate for the empirical covariance operators of a Hilbert-valued autoregressive process with random coefficients. Our approach is based on a suitable autoregressive representation of a sequence of covariance operators related to the model, which leads to a decomposition with Hilbert-valued martingale differences. Using large deviation inequalities for Hilbert-valued martingale differences, we then establish exponential bounds and derive the almost sure convergence of the empirical covariance operators in the Hilbert-Schmidt norm, achieving the parametric rate n up to a ln(n) factor in the bounded process case.