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Limit theorems for quadratic forms of Lévy-driven continuous-time linear processes

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  • Bai, Shuyang
  • Ginovyan, Mamikon S.
  • Taqqu, Murad S.

Abstract

We study the asymptotic behavior of a suitable normalized stochastic process {QT(t),t∈[0,1]}. This stochastic process is generated by a Toeplitz type quadratic functional of a Lévy-driven continuous-time linear process. We show that under some Lp-type conditions imposed on the covariance function of the model and the kernel of the quadratic functional, the process QT(t) obeys a central limit theorem, that is, the finite-dimensional distributions of the standard T normalized process QT(t) tend to those of a normalized standard Brownian motion. In contrast, when the covariance function of the model and the kernel of the quadratic functional have a slow power decay, then we have a non-central limit theorem for QT(t), that is, the finite-dimensional distributions of the process QT(t), normalized by Tγ for some γ>1/2, tend to those of a non-Gaussian non-stationary-increment self-similar process which can be represented by a double stochastic Wiener–Itô integral on R2.

Suggested Citation

  • Bai, Shuyang & Ginovyan, Mamikon S. & Taqqu, Murad S., 2016. "Limit theorems for quadratic forms of Lévy-driven continuous-time linear processes," Stochastic Processes and their Applications, Elsevier, vol. 126(4), pages 1036-1065.
    • Handle: RePEc:eee:spapps:v:126:y:2016:i:4:p:1036-1065
    DOI: 10.1016/j.spa.2015.10.010
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    References listed on IDEAS

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    1. Bai, Shuyang & Ginovyan, Mamikon S. & Taqqu, Murad S., 2015. "Functional limit theorems for Toeplitz quadratic functionals of continuous time Gaussian stationary processes," Statistics & Probability Letters, Elsevier, vol. 104(C), pages 58-67.
    2. P. Brockwell, 2001. "Lévy-Driven Carma Processes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 53(1), pages 113-124, March.
    3. P. Brockwell, 2014. "Recent results in the theory and applications of CARMA processes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 66(4), pages 647-685, August.
    4. Pipiras, Vladas & Taqqu, Murad S., 2010. "Regularization and integral representations of Hermite processes," Statistics & Probability Letters, Elsevier, vol. 80(23-24), pages 2014-2023, December.
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    Cited by:

    1. A. V. Ivanov & N. N. Leonenko & I. V. Orlovskyi, 2020. "On the Whittle estimator for linear random noise spectral density parameter in continuous-time nonlinear regression models," Statistical Inference for Stochastic Processes, Springer, vol. 23(1), pages 129-169, April.

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