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Spectral analysis of weakly stationary processes valued in a separable Hilbert space

Abstract : In this paper, we review and clarify the construction of a spectral theory for weakly-stationary processes valued in a separable Hilbert space. We introduce the basic fundamental concepts and results of functional analysis and operator theory needed to follow the way paved by Payen in [52], Mandrekar and Salehi in [45] and Kakihara in [33]. They lead us to view the spectral representation of a weakly stationary Hilbert valued time series as a gramian isometry between its time domain and its spectral domain. Time invariant linear filters with Hilbert-valued inputs and outputs are then defined through their operator transfer functions in the spectral domain. General results on the composition and inversion of such filters follow naturally. Spectral representations have enjoyed a renewed interest in the context of functional time series. The gramian isometry between the time and spectral domains constitutes an interesting and enlightening complement to recent approaches such as the one proposed in [50]. We also provide an overview of recent statistical results for the spectral analysis of functional time-series.
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Contributor : Amaury Durand Connect in order to contact the contributor
Submitted on : Monday, July 6, 2020 - 10:52:22 AM
Last modification on : Tuesday, October 19, 2021 - 11:16:16 AM


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  • HAL Id : hal-02318267, version 3


Amaury Durand, François Roueff. Spectral analysis of weakly stationary processes valued in a separable Hilbert space. 2020. ⟨hal-02318267v3⟩



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