Spectral analysis and parameter estimation of Gaussian functional time series
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This paper contributes with some asymptotic results to the spectral analysis of functional time series. Specifically, the convergence to zero, in the Hilbert-Schmidt operator norm, of the functional bias associated with the periodogram operator is obtained. The uniform convergence to zero of its eigenvalues then follows. The convergence to zero of its eigenvectors is also proved in a Hilbert space norm. Central and non-central limit results are formulated to derive the respective asymptotic probability distributions of the functional discrete Fourier transform (fDFT), and of the periodogram operator, in a Gaussian setting. Parameter estimation, based on the periodogram operator, is addressed. The strong–consistency of the formulated estimator is derived. The conditions assumed cover since the short–range (SRD) to the long–range (LRD) dependence Gaussian scenarios. Particularly, the memory parameter in LRD functional sequences can be estimated, as illustrated for the semiparametric family of fractional functional time series models introduced here.
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