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
Full-text PDF
Abstract |
References |
Similar Articles |
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
- Héctor Araya and Ciprian A. Tudor, Behavior of the Hermite sheet with respect to the Hurst index, Stochastic Process. Appl. 129 (2019), no. 7, 2582–2605. MR 3958443, DOI https://doi.org/10.1016/j.spa.2018.07.017
- Shuyang Bai and Murad S. Taqqu, Behavior of the generalized Rosenblatt process at extreme critical exponent values, Ann. Probab. 45 (2017), no. 2, 1278–1324. MR 3630299, DOI https://doi.org/10.1214/15-AOP1087
- Denis Bell and David Nualart, Noncentral limit theorem for the generalized Hermite process, Electron. Commun. Probab. 22 (2017), Paper No. 66, 13. MR 3734105, DOI https://doi.org/10.1214/17-ECP99
- Patrick Cheridito, Hideyuki Kawaguchi, and Makoto Maejima, Fractional Ornstein-Uhlenbeck processes, Electron. J. Probab. 8 (2003), no. 3, 14. MR 1961165, DOI https://doi.org/10.1214/EJP.v8-125
- Robert Fox and Murad S. Taqqu, Multiple stochastic integrals with dependent integrators, J. Multivariate Anal. 21 (1987), no. 1, 105–127. MR 877845, DOI https://doi.org/10.1016/0047-259X%2887%2990101-1
- Makoto Maejima and Ciprian 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, DOI https://doi.org/10.1080/07362990701540519
- Ivan Nourdin, Selected aspects of fractional Brownian motion, Bocconi & Springer Series, vol. 4, Springer, Milan; Bocconi University Press, Milan, 2012. MR 3076266
- Ivan Nourdin and Giovanni Peccati, Normal approximations with Malliavin calculus, Cambridge Tracts in Mathematics, vol. 192, Cambridge University Press, Cambridge, 2012. From Stein’s method to universality. MR 2962301
- David Nualart and Giovanni Peccati, Central limit theorems for sequences of multiple stochastic integrals, Ann. Probab. 33 (2005), no. 1, 177–193. MR 2118863, DOI https://doi.org/10.1214/009117904000000621
- Vladas Pipiras and Murad S. Taqqu, Long-range dependence and self-similarity, Cambridge Series in Statistical and Probabilistic Mathematics, [45], Cambridge University Press, Cambridge, 2017. MR 3729426
- Yu. V. Prokhorov, Convergence of random processes and limit theorems in probability theory, Teor. Veroyatnost. i Primenen. 1 (1956), 177–238 (Russian, with English summary). MR 0084896
- M. Slaoui and C. A. Tudor, On the Linear Stochastic Heat Equation with Hermite Noise, Preprint, 2017.
- Norma Terrin and Murad S. Taqqu, Power counting theorem in Euclidean space, Random walks, Brownian motion, and interacting particle systems, Progr. Probab., vol. 28, Birkhäuser Boston, Boston, MA, 1991, pp. 425–440. MR 1146462
- Ciprian A. Tudor, Analysis of variations for self-similar processes, Probability and its Applications (New York), Springer, Cham, 2013. A stochastic calculus approach. MR 3112799
- Mark S. Veillette and Murad S. Taqqu, Properties and numerical evaluation of the Rosenblatt distribution, Bernoulli 19 (2013), no. 3, 982–1005. MR 3079303, DOI https://doi.org/10.3150/12-BEJ421
- David Nualart, The Malliavin calculus and related topics, 2nd ed., Probability and its Applications (New York), Springer-Verlag, Berlin, 2006. MR 2200233
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
- D. Nualart and G. Peccati, Central limit theorems for sequences of multiple stochastic integrals, Ann. Probab. 33 (2005), 177–193. MR 2118863
- V. Pipiras and M. Taqqu, Long-range dependence and self-similarity, Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, 2017. MR 3729426
- Y. V. Prokhorov, Convergence of random processes and limit theorems in probability theory, Theory Probab. Appl. 1 (1956), no. 2, 157–214. MR 0084896
- M. Slaoui and C. A. Tudor, On the Linear Stochastic Heat Equation with Hermite Noise, Preprint, 2017.
- N. Terrin and M. S. Taqqu, Power counting theorem in Euclidean space, Random Walks, Brownian Motion, and Interacting Particle Systems, Progress in Probability, vol. 28, Birkhäuser, Boston, MA, 1991, pp. 425–440. MR 1146462
- C. A. Tudor, Analysis of Variations for Self-Similar Processes. A Stochastic Calculus Approach, Probability and its Applications, Springer, Cham, New York, 2013. MR 3112799
- M. S. Veillette and M. S. Taqqu, Properties and numerical evalution of the Rosenblatt process, Bernoulli 19 (2013), no. 3, 982–1005. MR 3079303
- D. Nualart, Malliavin Calculus and Related Topics, Second edition. Probability and its Applications (New York). Springer-Verlag, Berlin, 2006. MR 2200233
Similar Articles
Retrieve articles in Theory of Probability and Mathematical Statistics
with MSC (2010):
60H05,
60H15,
60G22
Retrieve articles in all journals
with MSC (2010):
60H05,
60H15,
60G22
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