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Identifying the spectral representation of Hilbertian time series

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  • Horta, Eduardo
  • Ziegelmann, Flavio

Abstract

We provide n-consistency results regarding estimation of the spectral representation of covariance operators of Hilbertian time series, in a setting with imperfect measurements. This is a generalization of the method developed in Bathia et al. (2010). The generalization relies on an important property of centered random elements in a separable Hilbert space, namely, that they lie almost surely in the closed linear span of the associated covariance operator. We provide a straightforward proof to this fact. This result is, to our knowledge, overlooked in the literature. It incidentally gives a rigorous formulation of Principal Component Analysis in Hilbert spaces.

Suggested Citation

  • Horta, Eduardo & Ziegelmann, Flavio, 2016. "Identifying the spectral representation of Hilbertian time series," Statistics & Probability Letters, Elsevier, vol. 118(C), pages 45-49.
    • Handle: RePEc:eee:stapro:v:118:y:2016:i:c:p:45-49
    DOI: 10.1016/j.spl.2016.06.014
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    References listed on IDEAS

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    1. Peter Hall & Céline Vial, 2006. "Assessing the finite dimensionality of functional data," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 68(4), pages 689-705, September.
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    1. Horta, Eduardo & Ziegelmann, Flavio, 2018. "Conjugate processes: Theory and application to risk forecasting," Stochastic Processes and their Applications, Elsevier, vol. 128(3), pages 727-755.
    2. Rodney V. Fonseca & Aluísio Pinheiro, 2020. "Wavelet estimation of the dimensionality of curve time series," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 72(5), pages 1175-1204, October.
    3. Horta, Eduardo & Ziegelmann, Flavio, 2018. "Dynamics of financial returns densities: A functional approach applied to the Bovespa intraday index," International Journal of Forecasting, Elsevier, vol. 34(1), pages 75-88.

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