Spectral analysis of long range dependence functional time series

This paper introduces a new modeling framework for the spectral analysis of long--range dependence (LRD) in functional sequences, beyond the usual structural modeling assumptions of the linear setting. Specifically, a semiparametric non--linear model is adopted in the functional spectral domain, involving a long--memory operator. We prove that this operator also characterizes the heavy--tail behavior, in time, of the inverse functional Fourier transform in the space of bounded linear operators. The non--summability in time of its trace norm then follows. Some particular cases are analyzed, including space varying fractionally integrated functional autoregressive moving averages processes. In the Gaussian case, a weak--consistent parametric estimator of the long--memory operator is obtained, by minimizing the operator norm of a divergence information based functional loss.