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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. 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.
    2. Peter J. Brockwell & Alexander Lindner, 2010. "Strictly stationary solutions of autoregressive moving average equations," Biometrika, Biometrika Trust, vol. 97(3), pages 765-772.
    3. 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.
    4. 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.
    5. 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).
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    Cited by:

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    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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