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Functional time series approach for inference on multifractional partial pseudodifferential equations

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FDE2 - Fractional differential equations

Long  Range Dependence (LRD) in functional sequences is characterized in the spectral domain under suitable conditions (see Ruiz-Medina,2021). Particularly,  the spectral representation of the discretesampling of the mean-square solution to multifractionalpseudodifferential equations can be approximated by tapering its continuous spectrum. A multifractionally integrated functionalautoregressive moving averages process (MIFAMA process) family is thenobtained.  For this  family, the  convergence to zero in theHilbert-Schmidt operator norm of the integrated bias of the periodogramoperator is proved. Under a Gaussian scenario, a  weak—consistentparametric estimator of the long—memory operator is then obtained byminimizing, in  the  norm of bounded linear operators, a divergenceinformation functional  loss. Application of these results to asymptoticinference on multifractional processes, including multifractionalBrownian motion, are  illustrated as well.REFERENCERuiz—Medina, M.D. (2021). Spectral analysis of long range dependence infunctional time series. ArXiv: 1912.07086v7.

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