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On the CLT for discrete Fourier transforms of functional time series

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  • Cerovecki, Clément
  • Hörmann, Siegfried

Abstract

The purpose of this paper is to derive sharp conditions for the asymptotic normality of a discrete Fourier transform of a functional time series (Xt:t≥1) defined, for all θ∈(−π,π], by Sn(θ)=Xte−iθ+⋯+Xte−inθ. Assuming that the function space is a Hilbert space we prove that a Central Limit Theorem (CLT) holds for almost all frequencies θ if the process (Xt) is stationary, ergodic and purely non-deterministic. Under slightly stronger assumptions we formulate versions which provide a CLT for fixed frequencies as well as for Sn(θn), when θn→θ0 is a sequence of fundamental frequencies. In particular we also deduce the regular CLT (θ=0) under new and very mild assumptions. We show that our results apply to the most commonly studied functional time series.

Suggested Citation

  • Cerovecki, Clément & Hörmann, Siegfried, 2017. "On the CLT for discrete Fourier transforms of functional time series," Journal of Multivariate Analysis, Elsevier, vol. 154(C), pages 282-295.
    • Handle: RePEc:eee:jmvana:v:154:y:2017:i:c:p:282-295
    DOI: 10.1016/j.jmva.2016.11.006
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    Cited by:

    1. Beare, Brendan K. & Seo, Won-Ki, 2020. "Representation Of I(1) And I(2) Autoregressive Hilbertian Processes," Econometric Theory, Cambridge University Press, vol. 36(5), pages 773-802, October.
    2. Cerovecki, Clément & Francq, Christian & Hörmann, Siegfried & Zakoïan, Jean-Michel, 2019. "Functional GARCH models: The quasi-likelihood approach and its applications," Journal of Econometrics, Elsevier, vol. 209(2), pages 353-375.
    3. Leucht, Anne & Paparoditis, Efstathios & Rademacher, Daniel & Sapatinas, Theofanis, 2022. "Testing equality of spectral density operators for functional processes," Journal of Multivariate Analysis, Elsevier, vol. 189(C).
    4. 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).
    5. Brendan K. Beare & Juwon Seo & Won-Ki Seo, 2017. "Cointegrated Linear Processes in Hilbert Space," Journal of Time Series Analysis, Wiley Blackwell, vol. 38(6), pages 1010-1027, November.
    6. Klepsch, J. & Klüppelberg, C., 2017. "An innovations algorithm for the prediction of functional linear processes," Journal of Multivariate Analysis, Elsevier, vol. 155(C), pages 252-271.
    7. Hörmann, Siegfried & Jammoul, Fatima, 2022. "Consistently recovering the signal from noisy functional data," Journal of Multivariate Analysis, Elsevier, vol. 189(C).
    8. van Delft, Anne, 2020. "A note on quadratic forms of stationary functional time series under mild conditions," Stochastic Processes and their Applications, Elsevier, vol. 130(7), pages 4206-4251.

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