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Strictly stationary solutions of ARMA equations in Banach spaces

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  • Spangenberg, Felix

Abstract

We obtain necessary and sufficient conditions for the existence of strictly stationary solutions of ARMA equations in Banach spaces with independent and identically distributed noise under certain assumptions. First, we obtain conditions for ARMA(1,q) equations by excluding zero and the unit circle from the spectrum of the operator of the AR part. We then extend this to ARMA(p,q) equations. Finally, we discuss various examples.

Suggested Citation

  • Spangenberg, Felix, 2013. "Strictly stationary solutions of ARMA equations in Banach spaces," Journal of Multivariate Analysis, Elsevier, vol. 121(C), pages 127-138.
    • Handle: RePEc:eee:jmvana:v:121:y:2013:i:c:p:127-138
    DOI: 10.1016/j.jmva.2013.06.007
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    References listed on IDEAS

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    1. Philippe C. Besse & Herve Cardot & David B. Stephenson, 2000. "Autoregressive Forecasting of Some Functional Climatic Variations," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 27(4), pages 673-687, December.
    2. Brockwell, Peter J. & Lindner, Alexander, 2009. "Existence and uniqueness of stationary Lévy-driven CARMA processes," Stochastic Processes and their Applications, Elsevier, vol. 119(8), pages 2660-2681, August.
    3. Meinguet, Thomas & Segers, Johan, 2010. "Regularly varying time series in Banach spaces," LIDAM Discussion Papers ISBA 2010002, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    4. Peter J. Brockwell & Alexander Lindner, 2010. "Strictly stationary solutions of autoregressive moving average equations," Biometrika, Biometrika Trust, vol. 97(3), pages 765-772.
    5. Peter Brockwell & Alexander Lindner & Bernd Vollenbröker, 2012. "Strictly stationary solutions of multivariate ARMA equations with i.i.d. noise," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 64(6), pages 1089-1119, December.
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    Cited by:

    1. Ruiz-Medina, María D. & Álvarez-Liébana, Javier, 2019. "Strongly consistent autoregressive predictors in abstract Banach spaces," Journal of Multivariate Analysis, Elsevier, vol. 170(C), pages 186-201.
    2. Chao Zhang & Piotr Kokoszka & Alexander Petersen, 2022. "Wasserstein autoregressive models for density time series," Journal of Time Series Analysis, Wiley Blackwell, vol. 43(1), pages 30-52, January.
    3. Klepsch, J. & Klüppelberg, C. & Wei, T., 2017. "Prediction of functional ARMA processes with an application to traffic data," Econometrics and Statistics, Elsevier, vol. 1(C), pages 128-149.
    4. Rice, Gregory & Wirjanto, Tony & Zhao, Yuqian, 2021. "Exploring volatility of crude oil intra-day return curves: a functional GARCH-X Model," MPRA Paper 109231, University Library of Munich, Germany.
    5. 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.

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