IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v128y2018i2p426-444.html
   My bibliography  Save this article

The Hausdorff dimension of multivariate operator-self-similar Gaussian random fields

Author

Listed:
  • Sönmez, Ercan

Abstract

Let {X(t):t∈Rd} be a multivariate operator-self-similar random field with values in Rm. Such fields were introduced in [22] and satisfy the scaling property {X(cEt):t∈Rd}=d{cDX(t):t∈Rd} for all c>0, where E is a d×d real matrix and D is an m×m real matrix. We solve an open problem in [22] by calculating the Hausdorff dimension of the range and graph of a trajectory over the unit cube K=[0,1]d in the Gaussian case. In particular, we enlighten the property that the Hausdorff dimension is determined by the real parts of the eigenvalues of E and D as well as the multiplicity of the eigenvalues of E and D.

Suggested Citation

  • Sönmez, Ercan, 2018. "The Hausdorff dimension of multivariate operator-self-similar Gaussian random fields," Stochastic Processes and their Applications, Elsevier, vol. 128(2), pages 426-444.
    • Handle: RePEc:eee:spapps:v:128:y:2018:i:2:p:426-444
    DOI: 10.1016/j.spa.2017.05.003
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414917301400
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2017.05.003?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    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. S. Davies & P. Hall, 1999. "Fractal analysis of surface roughness by using spatial data," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 61(1), pages 3-37.
    2. Biermé, Hermine & Meerschaert, Mark M. & Scheffler, Hans-Peter, 2007. "Operator scaling stable random fields," Stochastic Processes and their Applications, Elsevier, vol. 117(3), pages 312-332, March.
    3. Biermé, Hermine & Lacaux, Céline, 2009. "Hölder regularity for operator scaling stable random fields," Stochastic Processes and their Applications, Elsevier, vol. 119(7), pages 2222-2248, July.
    4. Li, Yuqiang & Xiao, Yimin, 2011. "Multivariate operator-self-similar random fields," Stochastic Processes and their Applications, Elsevier, vol. 121(6), pages 1178-1200, June.
    Full references (including those not matched with items on IDEAS)

    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. Didier, Gustavo & Meerschaert, Mark M. & Pipiras, Vladas, 2018. "Domain and range symmetries of operator fractional Brownian fields," Stochastic Processes and their Applications, Elsevier, vol. 128(1), pages 39-78.
    2. Kremer, D. & Scheffler, H.-P., 2019. "Operator-stable and operator-self-similar random fields," Stochastic Processes and their Applications, Elsevier, vol. 129(10), pages 4082-4107.
    3. Li, Yuqiang & Xiao, Yimin, 2011. "Multivariate operator-self-similar random fields," Stochastic Processes and their Applications, Elsevier, vol. 121(6), pages 1178-1200, June.
    4. Li, Yuqiang, 2011. "Fluctuation limits of site-dependent branching systems in critical and large dimensions," Statistics & Probability Letters, Elsevier, vol. 81(11), pages 1604-1611, November.
    5. Wu, Dongsheng & Xiao, Yimin, 2009. "Continuity in the Hurst index of the local times of anisotropic Gaussian random fields," Stochastic Processes and their Applications, Elsevier, vol. 119(6), pages 1823-1844, June.
    6. Kremer, D. & Scheffler, H.-P., 2020. "About atomless random measures on δ-rings," Statistics & Probability Letters, Elsevier, vol. 164(C).
    7. Biermé, Hermine & Lacaux, Céline & Scheffler, Hans-Peter, 2011. "Multi-operator scaling random fields," Stochastic Processes and their Applications, Elsevier, vol. 121(11), pages 2642-2677, November.
    8. Abry, Patrice & Didier, Gustavo, 2018. "Wavelet eigenvalue regression for n-variate operator fractional Brownian motion," Journal of Multivariate Analysis, Elsevier, vol. 168(C), pages 75-104.
    9. 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.
    10. Lim, C.Y. & Meerschaert, M.M. & Scheffler, H.-P., 2014. "Parameter estimation for operator scaling random fields," Journal of Multivariate Analysis, Elsevier, vol. 123(C), pages 172-183.
    11. Sun, Ying & Chang, Xiaohui & Guan, Yongtao, 2018. "Flexible and efficient estimating equations for variogram estimation," Computational Statistics & Data Analysis, Elsevier, vol. 122(C), pages 45-58.
    12. Myoungji Lee & Marc G. Genton & Mikyoung Jun, 2016. "Testing Self-Similarity Through Lamperti Transformations," Journal of Agricultural, Biological and Environmental Statistics, Springer;The International Biometric Society;American Statistical Association, vol. 21(3), pages 426-447, September.
    13. Panigrahi, Snigdha & Roy, Parthanil & Xiao, Yimin, 2021. "Maximal moments and uniform modulus of continuity for stable random fields," Stochastic Processes and their Applications, Elsevier, vol. 136(C), pages 92-124.
    14. Gneiting, Tilmann, 2002. "Compactly Supported Correlation Functions," Journal of Multivariate Analysis, Elsevier, vol. 83(2), pages 493-508, November.
    15. Lim, S.C. & Teo, L.P., 2009. "Gaussian fields and Gaussian sheets with generalized Cauchy covariance structure," Stochastic Processes and their Applications, Elsevier, vol. 119(4), pages 1325-1356, April.
    16. Hong, Yiping & Zhou, Zaiying & Yang, Ying, 2020. "Hypothesis testing for the smoothness parameter of Matérn covariance model on a regular grid," Journal of Multivariate Analysis, Elsevier, vol. 177(C).
    17. Pilipauskaitė, Vytautė & Surgailis, Donatas, 2017. "Scaling transition for nonlinear random fields with long-range dependence," Stochastic Processes and their Applications, Elsevier, vol. 127(8), pages 2751-2779.
    18. Kristoufek, Ladislav & Vosvrda, Miloslav, 2014. "Commodity futures and market efficiency," Energy Economics, Elsevier, vol. 42(C), pages 50-57.
    19. Ercan Sönmez, 2021. "Sample Path Properties of Generalized Random Sheets with Operator Scaling," Journal of Theoretical Probability, Springer, vol. 34(3), pages 1279-1298, September.
    20. Alina Bărbulescu & Cristian Ștefan Dumitriu, 2023. "Fractal Characterization of Brass Corrosion in Cavitation Field in Seawater," Sustainability, MDPI, vol. 15(4), pages 1-14, February.

    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:128:y:2018:i:2:p:426-444. 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.