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Weakly stationary stochastic processes valued in a separable Hilbert space: Gramian-Cramér representations and applications

Abstract : The spectral theory for weakly stationary processes valued in a separable Hilbert space has known renewed interest in the past decade. However, the recent literature on this topic is often based on restrictive assumptions or lacks important insights. In this paper, we follow earlier approaches which fully exploit the normal Hilbert module property of the space of Hilbert-valued random variables. This approach clarifies and completes the isomorphic relationship between the modular spectral domain to the modular time domain provided by the Gramian-Cramér representation. We also discuss the general Bochner theorem and provide useful results on the composition and inversion of lag-invariant linear filters. Finally, we derive the Cramér-Karhunen-Loève decomposition and harmonic functional principal component analysis without relying on simplifying assumptions.
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https://hal.telecom-paris.fr/hal-02318267
Contributor : Amaury Durand Connect in order to contact the contributor
Submitted on : Saturday, September 11, 2021 - 4:03:09 PM
Last modification on : Tuesday, October 19, 2021 - 11:16:31 AM

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  • HAL Id : hal-02318267, version 4

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Amaury Durand, François Roueff. Weakly stationary stochastic processes valued in a separable Hilbert space: Gramian-Cramér representations and applications. 2021. ⟨hal-02318267v4⟩

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