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Stationary Gaussian Markov processes as limits of stationary autoregressive time series

Author

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  • Ernst, Philip A.
  • Brown, Lawrence D.
  • Shepp, Larry
  • Wolpert, Robert L.

Abstract

We consider the class, Cp, of all zero mean stationary Gaussian processes, {Yt:t∈(−∞,∞)} with p derivatives, for which the vector valued process {(Yt(0),…,Yt(p)):t≥0} is a p+1-vector Markov process, where Yt(0)=Y(t). We provide a rigorous description and treatment of these stationary Gaussian processes as limits of stationary AR(p) time series.

Suggested Citation

  • Ernst, Philip A. & Brown, Lawrence D. & Shepp, Larry & Wolpert, Robert L., 2017. "Stationary Gaussian Markov processes as limits of stationary autoregressive time series," Journal of Multivariate Analysis, Elsevier, vol. 155(C), pages 180-186.
    • Handle: RePEc:eee:jmvana:v:155:y:2017:i:c:p:180-186
    DOI: 10.1016/j.jmva.2016.12.008
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    References listed on IDEAS

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    1. Duan, Jin-Chuan, 1997. "Augmented GARCH (p,q) process and its diffusion limit," Journal of Econometrics, Elsevier, vol. 79(1), pages 97-127, July.
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    3. 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.
    4. Peter J. Brockwell & Vincenzo Ferrazzano & Claudia Klüppelberg, 2013. "High-frequency sampling and kernel estimation for continuous-time moving average processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 34(3), pages 385-404, May.
    5. Peter J. Brockwell & Alexander Lindner, 2010. "Strictly stationary solutions of autoregressive moving average equations," Biometrika, Biometrika Trust, vol. 97(3), pages 765-772.
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    Cited by:

    1. Szarek, Dawid & Maraj-Zygmąt, Katarzyna & Sikora, Grzegorz & Krapf, Diego & Wyłomańska, Agnieszka, 2022. "Statistical test for anomalous diffusion based on empirical anomaly measure for Gaussian processes," Computational Statistics & Data Analysis, Elsevier, vol. 168(C).
    2. Douissi, Soukaina & Es-Sebaiy, Khalifa & Alshahrani, Fatimah & Viens, Frederi G., 2022. "AR(1) processes driven by second-chaos white noise: Berry–Esséen bounds for quadratic variation and parameter estimation," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 886-918.

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