Spectral analysis of multifractional LRD functional time series

Long Range Dependence (LRD) in functional sequences is characterized in the spectral domain under suitable conditions. Particularly, multifractionally integrated functional autoregressive moving averages processes 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. The results derived allow, in particular, to develop inference from the discrete sampling of the Gaussian solution to fractional and multifractional pseudodifferential models introduced in Anh et al. (Fract Calc Appl Anal 19(5):1161-1199, 2016; 19(6):1434–1459, 2016) and Kelbert (Adv Appl Probab 37(1):1–25, 2005).