LRD spectral analysis ofmultifractional functional time series on manifolds Ovalle Muñoz, Diana Paola Ruiz Medina, María Dolores Connected and compact two-point homogeneous spaces Ibragimov contrast function LRD multifractionally integrated functional time series Supplementary Information The online version contains supplementary material available at https://doi.org/10.1007/s11749-023-00913-7. This paper addresses the estimation of the second-order structure of a manifold crosstime random field (RF) displaying spatially varying Long Range Dependence (LRD), adopting the functional time series framework introduced in Ruiz-Medina (Fract Calc Appl Anal 25:1426–1458, 2022). Conditions for the asymptotic unbiasedness of the integrated periodogram operator in the Hilbert–Schmidt operator norm are derived beyond structural assumptions.Weak-consistent estimation of the long-memory operator is achieved under a semiparametric functional spectral framework in the Gaussian context. The case where the projected manifold process can display Short Range Dependence (SRD) and LRD at different manifold scales is also analyzed. The performance of both estimation procedures is illustrated in the simulation study, in the context of multifractionally integrated spherical functional autoregressive–moving average (SPHARMA(p,q)) processes. 2024-05-07T07:16:47Z 2024-05-07T07:16:47Z 2024-01-12 journal article Ovalle–Muñoz, D.P., Ruiz–Medina, M.D. LRD spectral analysis of multifractional functional time series on manifolds. TEST (2024). [https://doi.org/10.1007/s11749-023-00913-7] https://hdl.handle.net/10481/91458 10.1007/s11749-023-00913-7 eng http://creativecommons.org/licenses/by/4.0/ open access Atribución 4.0 Internacional Springer Nature