IDEAS home Printed from https://ideas.repec.org/a/spr/jotpro/v37y2024i2d10.1007_s10959-024-01316-6.html
   My bibliography  Save this article

Multivariate Random Fields Evolving Temporally Over Hyperbolic Spaces

Author

Listed:
  • Anatoliy Malyarenko

    (Mälardalen University)

  • Emilio Porcu

    (Research Center at Khalifa University
    ADIA Lab)

Abstract

Gaussian random fields are completely characterised by their mean value and covariance function. Random fields on hyperbolic spaces have been studied to a limited extent only, namely for the case of scalar-valued fields that are not evolving over time. This paper challenges the problem of the second-order characteristics of multivariate (vector-valued) random fields that evolve temporally over hyperbolic spaces. Specifically, we characterise the continuous space–time covariance functions that are isotropic (radially symmetric) over space (the hyperbolic space) and stationary over time (the real line). Our finding is the analogue of recent findings that have been shown for the case where the space is either the n-dimensional sphere or more generally a two-point homogeneous space. Our main result can be read as a spectral representation theorem, and we also detail the main result for the subcase of covariance functions having a spectrum that is absolutely continuous with respect to the Lebesgue measure (technical details are reported below).

Suggested Citation

  • Anatoliy Malyarenko & Emilio Porcu, 2024. "Multivariate Random Fields Evolving Temporally Over Hyperbolic Spaces," Journal of Theoretical Probability, Springer, vol. 37(2), pages 975-1000, June.
    • Handle: RePEc:spr:jotpro:v:37:y:2024:i:2:d:10.1007_s10959-024-01316-6
    DOI: 10.1007/s10959-024-01316-6
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10959-024-01316-6
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10959-024-01316-6?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. Emilio Porcu & Alfredo Alegria & Reinhard Furrer, 2018. "Modeling Temporally Evolving and Spatially Globally Dependent Data," International Statistical Review, International Statistical Institute, vol. 86(2), pages 344-377, August.
    2. Zuopeng Fu & Yizao Wang, 2020. "Stable Processes with Stationary Increments Parameterized by Metric Spaces," Journal of Theoretical Probability, Springer, vol. 33(3), pages 1737-1754, September.
    3. Bonfim, Rafaela N. & Menegatto, Valdir A., 2016. "Strict positive definiteness of multivariate covariance functions on compact two-point homogeneous spaces," Journal of Multivariate Analysis, Elsevier, vol. 152(C), pages 237-248.
    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. Cleanthous, Galatia & Georgiadis, Athanasios G. & Lang, Annika & Porcu, Emilio, 2020. "Regularity, continuity and approximation of isotropic Gaussian random fields on compact two-point homogeneous spaces," Stochastic Processes and their Applications, Elsevier, vol. 130(8), pages 4873-4891.
    2. Alessia Caponera, 2021. "SPHARMA approximations for stationary functional time series on the sphere," Statistical Inference for Stochastic Processes, Springer, vol. 24(3), pages 609-634, October.
    3. Bingham, N.H. & Symons, Tasmin L., 2019. "Dimension walks on Sd×R," Statistics & Probability Letters, Elsevier, vol. 147(C), pages 12-17.
    4. Tarik Faouzi & Emilio Porcu & Igor Kondrashuk & Moreno Bevilacqua, 2024. "Convergence arguments to bridge cauchy and matérn covariance functions," Statistical Papers, Springer, vol. 65(2), pages 645-660, April.
    5. Emilio Porcu & Philip A. White, 2022. "Random fields on the hypertorus: Covariance modeling and applications," Environmetrics, John Wiley & Sons, Ltd., vol. 33(1), February.
    6. Tingjin Chu & Jialuo Liu & Jun Zhu & Haonan Wang, 2022. "Spatio-Temporal Expanding Distance Asymptotic Framework for Locally Stationary Processes," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 84(2), pages 689-713, August.
    7. Emery, Xavier & Furrer, Reinhard & Porcu, Emilio, 2019. "A turning bands method for simulating isotropic Gaussian random fields on the sphere," Statistics & Probability Letters, Elsevier, vol. 144(C), pages 9-15.
    8. Laura M. Sangalli, 2021. "Spatial Regression With Partial Differential Equation Regularisation," International Statistical Review, International Statistical Institute, vol. 89(3), pages 505-531, December.
    9. Guella, Jean Carlo & Menegatto, Valdir Antonio & Porcu, Emilio, 2018. "Strictly positive definite multivariate covariance functions on spheres," Journal of Multivariate Analysis, Elsevier, vol. 166(C), pages 150-159.
    10. Liu, Jialuo & Chu, Tingjin & Zhu, Jun & Wang, Haonan, 2021. "Semiparametric method and theory for continuously indexed spatio-temporal processes," Journal of Multivariate Analysis, Elsevier, vol. 183(C).
    11. Bissiri, Pier Giovanni & Cleanthous, Galatia & Emery, Xavier & Nipoti, Bernardo & Porcu, Emilio, 2022. "Nonparametric Bayesian modelling of longitudinally integrated covariance functions on spheres," Computational Statistics & Data Analysis, Elsevier, vol. 176(C).
    12. Rachid Senoussi & Emilio Porcu, 2022. "Nonstationary space–time covariance functions induced by dynamical systems," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 49(1), pages 211-235, March.
    13. Chunsheng Ma, 2024. "Bifractional Brownian Motions on Metric Spaces," Journal of Theoretical Probability, Springer, vol. 37(2), pages 1299-1332, June.
    14. Estrade, Anne & Fariñas, Alessandra & Porcu, Emilio, 2019. "Covariance functions on spheres cross time: Beyond spatial isotropy and temporal stationarity," Statistics & Probability Letters, Elsevier, vol. 151(C), pages 1-7.

    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:spr:jotpro:v:37:y:2024:i:2:d:10.1007_s10959-024-01316-6. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    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.