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Gap between orthogonal projectors—Application to stationary processes

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  • Boudou, Alain
  • Viguier-Pla, Sylvie

Abstract

In the statistics of processes, the usual convergence of projectors does not provide a good idea of the continuity of some phenomena. In order to address it, we introduce a measure of the gap between two projectors, that we link to another type of convergence of sequences of projectors. This allows to define the gap between two spectral measures and then to develop various properties associated with the latter. In particular, we establish the continuity of the convolution product. A study of the closeness between two continuous random functions and the exhibition of a common filter for them illustrate the possible applications in functional statistics.

Suggested Citation

  • Boudou, Alain & Viguier-Pla, Sylvie, 2016. "Gap between orthogonal projectors—Application to stationary processes," Journal of Multivariate Analysis, Elsevier, vol. 146(C), pages 282-300.
    • Handle: RePEc:eee:jmvana:v:146:y:2016:i:c:p:282-300
    DOI: 10.1016/j.jmva.2015.10.002
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    References listed on IDEAS

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    1. Masry, Elias, 2005. "Nonparametric regression estimation for dependent functional data: asymptotic normality," Stochastic Processes and their Applications, Elsevier, vol. 115(1), pages 155-177, January.
    2. Boudou, Alain & Romain, Yves, 2002. "On spectral and random measures associated to discrete and continuous-time processes," Statistics & Probability Letters, Elsevier, vol. 59(2), pages 145-157, September.
    3. Horváth, Lajos & Rice, Gregory, 2015. "Testing for independence between functional time series," Journal of Econometrics, Elsevier, vol. 189(2), pages 371-382.
    4. Dauxois, J. & Pousse, A. & Romain, Y., 1982. "Asymptotic theory for the principal component analysis of a vector random function: Some applications to statistical inference," Journal of Multivariate Analysis, Elsevier, vol. 12(1), pages 136-154, March.
    5. Benhenni, K. & Hedli-Griche, S. & Rachdi, M. & Vieu, P., 2008. "Consistency of the regression estimator with functional data under long memory conditions," Statistics & Probability Letters, Elsevier, vol. 78(8), pages 1043-1049, June.
    6. Aneiros-Pérez, Germán & Vieu, Philippe, 2008. "Nonparametric time series prediction: A semi-functional partial linear modeling," Journal of Multivariate Analysis, Elsevier, vol. 99(5), pages 834-857, May.
    7. Boudou, Alain & Viguier-Pla, Sylvie, 2010. "Relation between unit operators proximity and their associated spectral measures," Statistics & Probability Letters, Elsevier, vol. 80(23-24), pages 1724-1732, December.
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    Cited by:

    1. Boudou, Alain & Viguier-Pla, Sylvie, 2019. "Commuter of operators in a Hilbert space," Journal of Multivariate Analysis, Elsevier, vol. 170(C), pages 244-262.
    2. Aneiros, Germán & Cao, Ricardo & Fraiman, Ricardo & Genest, Christian & Vieu, Philippe, 2019. "Recent advances in functional data analysis and high-dimensional statistics," Journal of Multivariate Analysis, Elsevier, vol. 170(C), pages 3-9.

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