Testing LRD in the spectral domain for functional time series in manifolds

A statistical hypothesis test for long range dependence (LRD) in manifold-supported functional time series is formulated in the spectral domain. The proposed test statistic operator is based on the weighted periodogram operator. It is assumed that the elements of the spectral density operator family are invariant with respect to the group of isometries of the manifold. A Central Limit Theorem is derived to obtain the asymptotic Gaussian distribution of the proposed test statistics operator under the null hypothesis. The rate of convergence to zero, in the Hilbert--Schmidt operator norm, of the bias of the integrated empirical second and fourth order cumulant spectral density operators is established under the alternative hypothesis. The consistency of the test is derived, from the consistency, in the sense of the integrated mean square error, of the weighted periodogram operator under LRD. Our proposal to implement, in practice, the testing approach is based on the temporal-frequency-varying Karhunen-Lo\'eve expansion obtained here for invariant random Hilbert-Schmidt kernels on manifolds. A simulation study illustrates the main results regarding asymptotic normality and consistency, and the empirical size and power properties of the proposed testing approach.