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A note on Herglotz’s theorem for time series on function spaces

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  • van Delft, Anne
  • Eichler, Michael

Abstract

In this article, we prove Herglotz’s theorem for Hilbert-valued time series. This requires the notion of an operator-valued measure, which we shall make precise for our setting. Herglotz’s theorem for functional time series allows to generalize existing results that are central to frequency domain analysis on the function space. In particular, we use this result to prove the existence of a functional Cramér representation of a large class of processes, including those with jumps in the spectral distribution and long-memory processes. We furthermore obtain an optimal finite dimensional reduction of the time series under weaker assumptions than available in the literature. The results of this paper therefore enable Fourier analysis for processes of which the spectral density operator does not necessarily exist.

Suggested Citation

  • van Delft, Anne & Eichler, Michael, 2020. "A note on Herglotz’s theorem for time series on function spaces," Stochastic Processes and their Applications, Elsevier, vol. 130(6), pages 3687-3710.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:6:p:3687-3710
    DOI: 10.1016/j.spa.2019.10.006
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    References listed on IDEAS

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    1. McElroy, Tucker S. & Politis, Dimitris N., 2014. "Spectral density and spectral distribution inference for long memory time series via fixed-b asymptotics," Journal of Econometrics, Elsevier, vol. 182(1), pages 211-225.
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    4. Panaretos, Victor M. & Tavakoli, Shahin, 2013. "Cramér–Karhunen–Loève representation and harmonic principal component analysis of functional time series," Stochastic Processes and their Applications, Elsevier, vol. 123(7), pages 2779-2807.
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    Cited by:

    1. Alessia Caponera, 2021. "SPHARMA approximations for stationary functional time series on the sphere," Statistical Inference for Stochastic Processes, Springer, vol. 24(3), pages 609-634, October.
    2. Rademacher, Daniel & Kreiß, Jens-Peter & Paparoditis, Efstathios, 2024. "Asymptotic normality of spectral means of Hilbert space valued random processes," Stochastic Processes and their Applications, Elsevier, vol. 173(C).

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