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Regularization and integral representations of Hermite processes

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  • Pipiras, Vladas
  • Taqqu, Murad S.

Abstract

It is known that Hermite processes have a finite-time interval representation. For fractional Brownian motion, the representation has been well known and plays a fundamental role in developing stochastic calculus for the process. For the Rosenblatt process, the finite-time interval representation was originally established by using cumulants. The representation was extended to general Hermite processes through the convergence of suitable partial sum processes. We provide here an alternative and different proof for the finite-time interval representation of Hermite processes. The approach is based on regularization of Hermite processes and the fractional Gaussian noises underlying them, and does not use cumulants nor convergence of partial sums.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:stapro:v:80:y:2010:i:23-24:p:2014-2023
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    References listed on IDEAS

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    1. Albin, J. M. P., 1998. "A note on Rosenblatt distributions," Statistics & Probability Letters, Elsevier, vol. 40(1), pages 83-91, September.
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    Cited by:

    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. Maejima, Makoto & Tudor, Ciprian A., 2013. "On the distribution of the Rosenblatt process," Statistics & Probability Letters, Elsevier, vol. 83(6), pages 1490-1495.
    3. 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.
    4. Bai, Shuyang & Taqqu, Murad S., 2014. "Generalized Hermite processes, discrete chaos and limit theorems," Stochastic Processes and their Applications, Elsevier, vol. 124(4), pages 1710-1739.
    5. Bai, Shuyang & Taqqu, Murad S., 2013. "Multivariate limits of multilinear polynomial-form processes with long memory," Statistics & Probability Letters, Elsevier, vol. 83(11), pages 2473-2485.
    6. Shuyang Bai & Murad S. Taqqu, 2013. "Multivariate Limit Theorems In The Context Of Long-Range Dependence," Journal of Time Series Analysis, Wiley Blackwell, vol. 34(6), pages 717-743, November.
    7. Stoyan V. Stoyanov & Svetlozar T. Rachev & Stefan Mittnik & Frank J. Fabozzi, 2019. "Pricing Derivatives In Hermite Markets," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(06), pages 1-27, September.
    8. Bardet, Jean-Marc & Tudor, Ciprian, 2014. "Asymptotic behavior of the Whittle estimator for the increments of a Rosenblatt process," Journal of Multivariate Analysis, Elsevier, vol. 131(C), pages 1-16.

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