Normalité asymptotique locale de processus autorégressifs hilbertiens

We give a Local Asymptotic Normality (LAN condition) and a Uniform Local Asymptotic Normality (ULAN condition) for a class of Hilbert space valued autoregressive processes when the correlation operator depends upon an unknown one-dimensional parameter. We then derive a Hajek minimax bound, the consistency, the asymptotic normality, and the efficiency of the conditional maximum likelihood and Bayes estimators yielding their optimality.