On local asymptotic normality for functional autoregressive processes

We establish Local Asymptotic Normality (LAN) and Uniform Local Asymptotic Normality (ULAN) conditions for a class of function space valued autoregressive processes when the correlation operator depends on an unknown one-dimensional parameter. We then derive Hajek minimax bound, consistency, asymptotic normality and efficiency of the conditional maximum likelihood estimator yielding their optimality. A simulation studies illustrate the performance of the estimators.