Spectral analysis of long range dependence functional time series

Long Range Dependence (LRD) in functional sequences is characterized in the spectral domain under suitable conditions. Particularly, fractionally integrated functional autoregressive moving averages processes of variable order can be introduced in this framework. The convergence to zero in the Hilbert-Schmidt operator norm of the integrated bias of the periodogram operator is proved. Under a Gaussian scenario, a weak--consistent parametric estimator of the long--memory operator is then obtained by minimizing, in the norm of bounded linear operators, a divergence information functional loss.