LRD spectral analysis of multifractional functional time series on manifolds

This paper introduces the statistical analysis of Jacobi frequency varying Long Range Dependence (LRD) functional time series in connected and compact two-point homogeneous spaces. The convergence to zero, in the Hilbert-Schmidt operator norm, of the integrated bias of the periodogram operator is proved under alternative conditions to the ones considered in Ruiz-Medina (2022). Under this setting of conditions, weak-consistency of the minimum contrast parameter estimator of the LRD operator holds. The case where the projected manifold process can display Short Range Dependence (SRD) and LRD at different manifold scales is also analyzed. The estimation of the spectral density operator is addressed in this case. The performance of both estimation procedures is illustrated in the simulation study undertaken within the families of multifractionally integrated spherical functional autoregressive-moving average (SPHARMA) processes.