Limit behavior of the Rosenblatt Ornstein–Uhlenbeck process with respect to the Hurst index
Authors:
M. Slaoui and C. A. Tudor
Journal:
Theor. Probability and Math. Statist. 98 (2019), 183-198
MSC (2010):
Primary 60H05, 60H15, 60G22
DOI:
https://doi.org/10.1090/tpms/1070
Published electronically:
August 19, 2019
MathSciNet review:
3824686
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Additional Information
Abstract: We study the convergence in distribution, as $H\to \frac {1}{2}$ and as $H\to 1$, of the integral $\int _{\mathbb {R}} f(u) dZ^{H}(u)$, where $Z^{H}$ is a Rosenblatt process with self-similarity index $H\in \left (\frac {1}{2}, 1\right )$ and $f$ is a suitable deterministic function. We focus our analysis on the case of the Rosenblatt Ornstein–Uhlenbeck process, which is the solution of the Langevin equation driven by the Rosenblatt process.
References
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References
- H. Araya and C. A. Tudor, Behavior of the Hermite Sheet with Respect to the Hurst Index, Stochastic Process. Appl. 129 (2019), 2582–2605. MR 3958443
- S. Bai and M. Taqqu, Behavior of the generalized Rosenblatt process at extremes critical exponent values, Ann. Probab. 45 (2017), no. 2, 1278–1324. MR 3630299
- D. Bell and D. Nualart, Noncentral limit theorem for the generalized Rosenblatt process, Electron. Commun. Probab. 22 (2017), no. 66, 13. MR 3734105
- P. Cheridito, H. Kawaguchi, and M. Maejima, Fractional Ornstein–Uhlenbeck processes, Electron. J. Probab. 8 (2003), no. 3, 1–14. MR 1961165
- R. Fox and M. S. Taqqu, Multiple stochastic integrals with dependent integrators, J. Multivariate Anal. 21 (1987), 105–127. MR 877845
- M. Maejima and C. A. Tudor, Wiener integrals with respect to the Hermite process and a Non-Central Limit Theorem, Stoch. Anal. Appl. 25 (2007), no. 5, 1043–1056. MR 2352951
- I. Nourdin, Selected Aspects of the Fractional Brownian Motion, Springer-Bocconi, 2012. MR 3076266
- I. Nourdin and G. Peccati, Normal Approximations with Malliavin Calculus From Stein’s Method to Universality, Cambridge University Press, 2012. MR 2962301
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Additional Information
M. Slaoui
Affiliation:
Laboratoire Paul Painlevé, Université de Lille 1, F-59655 Villeneuve d’Ascq, France
Email:
meryem.slaoui@math.univ-lille1.fr
C. A. Tudor
Affiliation:
Laboratoire Paul Painlevé, Université de Lille 1, F-59655 Villeneuve d’Ascq, France, and ISMMA, Romanian Academy, Romania
Email:
tudor@math.univ-lille1.fr
Keywords:
Wiener chaos,
Rosenblatt process,
cumulants,
Hurst parameter
Received by editor(s):
January 8, 2018
Published electronically:
August 19, 2019
Article copyright:
© Copyright 2019
American Mathematical Society
