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Asymptotic normality of the principal components of functional time series

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  • Kokoszka, Piotr
  • Reimherr, Matthew

Abstract

We establish the asymptotic normality of the sample principal components of functional stochastic processes under nonrestrictive assumptions which admit nonlinear functional time series models. We show that the aforementioned asymptotic depends only on the asymptotic normality of the sample covariance operator, and that the latter condition holds for weakly dependent functional time series which admit expansions as Bernoulli shifts. The weak dependence is quantified by the condition of L4-m-approximability which includes all functional time series models in practical use. We also demonstrate convergence of the cross covariance operators of the sample functional principal components to their counterparts in the normal limit.

Suggested Citation

  • Kokoszka, Piotr & Reimherr, Matthew, 2013. "Asymptotic normality of the principal components of functional time series," Stochastic Processes and their Applications, Elsevier, vol. 123(5), pages 1546-1562.
    • Handle: RePEc:eee:spapps:v:123:y:2013:i:5:p:1546-1562
    DOI: 10.1016/j.spa.2012.12.011
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    Cited by:

    1. Reimherr, Matthew, 2015. "Functional regression with repeated eigenvalues," Statistics & Probability Letters, Elsevier, vol. 107(C), pages 62-70.
    2. Kokoszka, Piotr & Kulik, Rafał, 2023. "Principal component analysis of infinite variance functional data," Journal of Multivariate Analysis, Elsevier, vol. 193(C).
    3. 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.
    4. Berkes, István & Horváth, Lajos & Rice, Gregory, 2016. "On the asymptotic normality of kernel estimators of the long run covariance of functional time series," Journal of Multivariate Analysis, Elsevier, vol. 144(C), pages 150-175.
    5. Petrovich, Justin & Reimherr, Matthew, 2017. "Asymptotic properties of principal component projections with repeated eigenvalues," Statistics & Probability Letters, Elsevier, vol. 130(C), pages 42-48.
    6. Cai, Leheng & Hu, Qirui, 2024. "Simultaneous inference and uniform test for eigensystems of functional data," Computational Statistics & Data Analysis, Elsevier, vol. 192(C).
    7. Kraus, David, 2019. "Inferential procedures for partially observed functional data," Journal of Multivariate Analysis, Elsevier, vol. 173(C), pages 583-603.
    8. Lakraj, Gamage Pemantha & Ruymgaart, Frits, 2017. "Some asymptotic theory for Silverman’s smoothed functional principal components in an abstract Hilbert space," Journal of Multivariate Analysis, Elsevier, vol. 155(C), pages 122-132.
    9. 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).
    10. Dimitrios Pilavakis & Efstathios Paparoditis & Theofanis Sapatinas, 2020. "Testing equality of autocovariance operators for functional time series," Journal of Time Series Analysis, Wiley Blackwell, vol. 41(4), pages 571-589, July.

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