IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v120y2010i12p2331-2362.html
   My bibliography  Save this article

A wavelet analysis of the Rosenblatt process: Chaos expansion and estimation of the self-similarity parameter

Author

Listed:
  • Bardet, J.-M.
  • Tudor, C.A.

Abstract

By using chaos expansion into multiple stochastic integrals, we make a wavelet analysis of two self-similar stochastic processes: the fractional Brownian motion and the Rosenblatt process. We study the asymptotic behavior of the statistic based on the wavelet coefficients of these processes. Basically, when applied to a non-Gaussian process (such as the Rosenblatt process) this statistic satisfies a non-central limit theorem even when we increase the number of vanishing moments of the wavelet function. We apply our limit theorems to construct estimators for the self-similarity index and we illustrate our results by simulations.

Suggested Citation

  • Bardet, J.-M. & Tudor, C.A., 2010. "A wavelet analysis of the Rosenblatt process: Chaos expansion and estimation of the self-similarity parameter," Stochastic Processes and their Applications, Elsevier, vol. 120(12), pages 2331-2362, December.
    • Handle: RePEc:eee:spapps:v:120:y:2010:i:12:p:2331-2362
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304-4149(10)00189-4
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. F. Roueff & M. S. Taqqu, 2009. "Asymptotic normality of wavelet estimators of the memory parameter for linear processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 30(5), pages 534-558, September.
    2. Jean‐Marc Bardet & Pierre R. Bertrand, 2010. "A Non‐Parametric Estimator of the Spectral Density of a Continuous‐Time Gaussian Process Observed at Random Times," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 37(3), pages 458-476, September.
    3. Fox, Robert & Taqqu, Murad S., 1987. "Multiple stochastic integrals with dependent integrators," Journal of Multivariate Analysis, Elsevier, vol. 21(1), pages 105-127, February.
    4. Nualart, D. & Ortiz-Latorre, S., 2008. "Central limit theorems for multiple stochastic integrals and Malliavin calculus," Stochastic Processes and their Applications, Elsevier, vol. 118(4), pages 614-628, April.
    5. Taqqu, Murad S., 1978. "A representation for self-similar processes," Stochastic Processes and their Applications, Elsevier, vol. 7(1), pages 55-64, March.
    6. Breuer, Péter & Major, Péter, 1983. "Central limit theorems for non-linear functionals of Gaussian fields," Journal of Multivariate Analysis, Elsevier, vol. 13(3), pages 425-441, September.
    7. J. Bardet & G. Lang & E. Moulines & P. Soulier, 2000. "Wavelet Estimator of Long-Range Dependent Processes," Statistical Inference for Stochastic Processes, Springer, vol. 3(1), pages 85-99, January.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Daw, Lara & Kerchev, George, 2023. "Fractal dimensions of the Rosenblatt process," Stochastic Processes and their Applications, Elsevier, vol. 161(C), pages 544-571.
    2. Patrice Abry & Gustavo Didier & Hui Li, 2019. "Two-step wavelet-based estimation for Gaussian mixed fractional processes," Statistical Inference for Stochastic Processes, Springer, vol. 22(2), pages 157-185, July.
    3. Yuriy Kozachenko & Andriy Olenko & Olga Polosmak, 2015. "Convergence in L p ([0, T]) of Wavelet Expansions of φ-Sub-Gaussian Random Processes," Methodology and Computing in Applied Probability, Springer, vol. 17(1), pages 139-153, March.
    4. Čoupek, Petr & Duncan, Tyrone E. & Pasik-Duncan, Bozenna, 2022. "A stochastic calculus for Rosenblatt processes," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 853-885.
    5. Xichao Sun & Litan Yan & Yong Ge, 2022. "The Laws of Large Numbers Associated with the Linear Self-attracting Diffusion Driven by Fractional Brownian Motion and Applications," Journal of Theoretical Probability, Springer, vol. 35(3), pages 1423-1478, September.
    6. Obayda Assaad & Ciprian A. Tudor, 2020. "Parameter identification for the Hermite Ornstein–Uhlenbeck process," Statistical Inference for Stochastic Processes, Springer, vol. 23(2), pages 251-270, July.
    7. Ayache, Antoine, 2020. "Lower bound for local oscillations of Hermite processes," Stochastic Processes and their Applications, Elsevier, vol. 130(8), pages 4593-4607.
    8. 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.
    9. Petr Čoupek & Viktor Dolník & Zdeněk Hlávka & Daniel Hlubinka, 2024. "Fourier approach to goodness-of-fit tests for Gaussian random processes," Statistical Papers, Springer, vol. 65(5), pages 2937-2972, July.
    10. 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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. 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.
    2. Surgailis, Donatas & Teyssière, Gilles & Vaiciulis, Marijus, 2008. "The increment ratio statistic," Journal of Multivariate Analysis, Elsevier, vol. 99(3), pages 510-541, March.
    3. Mikko S. Pakkanen & Anthony Réveillac, 2014. "Functional limit theorems for generalized variations of the fractional Brownian sheet," CREATES Research Papers 2014-14, Department of Economics and Business Economics, Aarhus University.
    4. Harnett, Daniel & Nualart, David, 2018. "Central limit theorem for functionals of a generalized self-similar Gaussian process," Stochastic Processes and their Applications, Elsevier, vol. 128(2), pages 404-425.
    5. Noreddine, Salim & Nourdin, Ivan, 2011. "On the Gaussian approximation of vector-valued multiple integrals," Journal of Multivariate Analysis, Elsevier, vol. 102(6), pages 1008-1017, July.
    6. José Manuel Corcuera, 2012. "New Central Limit Theorems for Functionals of Gaussian Processes and their Applications," Methodology and Computing in Applied Probability, Springer, vol. 14(3), pages 477-500, September.
    7. Anh, V. V. & Leonenko, N. N., 1999. "Non-Gaussian scenarios for the heat equation with singular initial conditions," Stochastic Processes and their Applications, Elsevier, vol. 84(1), pages 91-114, November.
    8. Nikolai Leonenko & Ludmila Sakhno, 2001. "On the Kaplan–Meier Estimator of Long-Range Dependent Sequences," Statistical Inference for Stochastic Processes, Springer, vol. 4(1), pages 17-40, January.
    9. Kerstin Gärtner & Mark Podolskij, 2014. "On non-standard limits of Brownian semi-stationary," CREATES Research Papers 2014-50, Department of Economics and Business Economics, Aarhus University.
    10. Muniandy, Sithi V. & Uning, Rosemary, 2006. "Characterization of exchange rate regimes based on scaling and correlation properties of volatility for ASEAN-5 countries," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 371(2), pages 585-598.
    11. Chi, Zhiyi, 2001. "Stationary self-similar random fields on the integer lattice," Stochastic Processes and their Applications, Elsevier, vol. 91(1), pages 99-113, January.
    12. Bai, Shuyang & Taqqu, Murad S. & Zhang, Ting, 2016. "A unified approach to self-normalized block sampling," Stochastic Processes and their Applications, Elsevier, vol. 126(8), pages 2465-2493.
    13. Hu, Yaozhong & Nualart, David, 2010. "Parameter estimation for fractional Ornstein-Uhlenbeck processes," Statistics & Probability Letters, Elsevier, vol. 80(11-12), pages 1030-1038, June.
    14. Eden, Richard & Víquez, Juan, 2015. "Nourdin–Peccati analysis on Wiener and Wiener–Poisson space for general distributions," Stochastic Processes and their Applications, Elsevier, vol. 125(1), pages 182-216.
    15. 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.
    16. Yoon-Tae Kim & Hyun-Suk Park, 2022. "Fourth Cumulant Bound of Multivariate Normal Approximation on General Functionals of Gaussian Fields," Mathematics, MDPI, vol. 10(8), pages 1-17, April.
    17. Matthieu Garcin & Dominique Guegan, 2015. "Optimal wavelet shrinkage of a noisy dynamical system with non-linear noise impact," Documents de travail du Centre d'Economie de la Sorbonne 15085, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    18. Barndorff-Nielsen, Ole E. & Corcuera, José Manuel & Podolskij, Mark, 2009. "Power variation for Gaussian processes with stationary increments," Stochastic Processes and their Applications, Elsevier, vol. 119(6), pages 1845-1865, June.
    19. Miguel A. Arcones, 1999. "The Law of the Iterated Logarithm over a Stationary Gaussian Sequence of Random Vectors," Journal of Theoretical Probability, Springer, vol. 12(3), pages 615-641, July.
    20. Jean-Christophe Breton & Jean-François Coeurjolly, 2012. "Confidence intervals for the Hurst parameter of a fractional Brownian motion based on finite sample size," Statistical Inference for Stochastic Processes, Springer, vol. 15(1), pages 1-26, April.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:spapps:v:120:y:2010:i:12:p:2331-2362. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.