IDEAS home Printed from https://ideas.repec.org/a/eee/stapro/v171y2021ics0167715221000031.html
   My bibliography  Save this article

On Lévy’s Brownian motion and white noise space on the circle

Author

Listed:
  • Huang, Chunfeng
  • Li, Ao

Abstract

In this article, we show that the Brownian motion on the circle constructed by Lévy (1959) is a regular Euclidean Brownian motion on the half-circle with its own mirror image on the other half-circle, and is degenerated in the sense of Minlos (1959). This raises the question of what the white noise is on the circle. We then formally define the white noise space and its associated Brownian bridge on the circle.

Suggested Citation

  • Huang, Chunfeng & Li, Ao, 2021. "On Lévy’s Brownian motion and white noise space on the circle," Statistics & Probability Letters, Elsevier, vol. 171(C).
    • Handle: RePEc:eee:stapro:v:171:y:2021:i:c:s0167715221000031
    DOI: 10.1016/j.spl.2021.109041
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167715221000031
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spl.2021.109041?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

    for a different version of it.

    References listed on IDEAS

    as
    1. Wood, Andrew T. A., 1995. "When is a truncated covariance function on the line a covariance function on the circle?," Statistics & Probability Letters, Elsevier, vol. 24(2), pages 157-164, August.
    2. Finn Lindgren & Håvard Rue & Johan Lindström, 2011. "An explicit link between Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 73(4), pages 423-498, September.
    3. Giacomo Aletti & Matteo Ruffini, 2017. "Is the Brownian bridge a good noise model on the boundary of a circle?," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 69(2), pages 389-416, April.
    4. Asger Hobolth & Jan Pedersen & Eva Jensen, 2003. "A continuous parametric shape model," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 55(2), pages 227-242, June.
    5. Huang, Chunfeng & Zhang, Haimeng & Robeson, Scott M., 2016. "Intrinsic random functions and universal kriging on the circle," Statistics & Probability Letters, Elsevier, vol. 108(C), pages 33-39.
    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. Rajala, T. & Penttinen, A., 2014. "Bayesian analysis of a Gibbs hard-core point pattern model with varying repulsion range," Computational Statistics & Data Analysis, Elsevier, vol. 71(C), pages 530-541.
    2. K. Shuvo Bakar & Nicholas Biddle & Philip Kokic & Huidong Jin, 2020. "A Bayesian spatial categorical model for prediction to overlapping geographical areas in sample surveys," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 183(2), pages 535-563, February.
    3. Laura M. Sangalli, 2021. "Spatial Regression With Partial Differential Equation Regularisation," International Statistical Review, International Statistical Institute, vol. 89(3), pages 505-531, December.
    4. Matthias Katzfuss & Joseph Guinness & Wenlong Gong & Daniel Zilber, 2020. "Vecchia Approximations of Gaussian-Process Predictions," Journal of Agricultural, Biological and Environmental Statistics, Springer;The International Biometric Society;American Statistical Association, vol. 25(3), pages 383-414, September.
    5. Finn Lindgren, 2015. "Comments on: Comparing and selecting spatial predictors using local criteria," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 24(1), pages 35-44, March.
    6. Andre Python & Andreas Bender & Marta Blangiardo & Janine B. Illian & Ying Lin & Baoli Liu & Tim C.D. Lucas & Siwei Tan & Yingying Wen & Davit Svanidze & Jianwei Yin, 2022. "A downscaling approach to compare COVID‐19 count data from databases aggregated at different spatial scales," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 185(1), pages 202-218, January.
    7. Ruiman Zhong & Paula Moraga, 2024. "Bayesian Hierarchical Models for the Combination of Spatially Misaligned Data: A Comparison of Melding and Downscaler Approaches Using INLA and SPDE," Journal of Agricultural, Biological and Environmental Statistics, Springer;The International Biometric Society;American Statistical Association, vol. 29(1), pages 110-129, March.
    8. Daniela Silva & Raquel Menezes & Ana Moreno & Ana Teles-Machado & Susana Garrido, 2024. "Environmental Effects on the Spatiotemporal Variability of Sardine Distribution Along the Portuguese Continental Coast," Journal of Agricultural, Biological and Environmental Statistics, Springer;The International Biometric Society;American Statistical Association, vol. 29(3), pages 553-575, September.
    9. Vanhatalo, Jarno & Veneranta, Lari & Hudd, Richard, 2012. "Species distribution modeling with Gaussian processes: A case study with the youngest stages of sea spawning whitefish (Coregonus lavaretus L. s.l.) larvae," Ecological Modelling, Elsevier, vol. 228(C), pages 49-58.
    10. Xiaoyu Xiong & Benjamin D. Youngman & Theodoros Economou, 2021. "Data fusion with Gaussian processes for estimation of environmental hazard events," Environmetrics, John Wiley & Sons, Ltd., vol. 32(3), May.
    11. Kirchner, Kristin & Willems, Joshua, 2025. "Multiple and weak Markov properties in Hilbert spaces with applications to fractional stochastic evolution equations," Stochastic Processes and their Applications, Elsevier, vol. 186(C).
    12. Yuan Yan & Eva Cantoni & Chris Field & Margaret Treble & Joanna Mills Flemming, 2023. "Spatiotemporal modeling of mature‐at‐length data using a sliding window approach," Environmetrics, John Wiley & Sons, Ltd., vol. 34(2), March.
    13. Jonathan Rougier & Andrew Zammit-Mangion, 2016. "Visualization for Large-scale Gaussian Updates," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 43(4), pages 1153-1161, December.
    14. Marcelo Cunha & Dani Gamerman & Montserrat Fuentes & Marina Paez, 2017. "A non-stationary spatial model for temperature interpolation applied to the state of Rio de Janeiro," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 66(5), pages 919-939, November.
    15. Caamaño-Carrillo, Christian & Bevilacqua, Moreno & López, Cristian & Morales-Oñate, Víctor, 2024. "Nearest neighbors weighted composite likelihood based on pairs for (non-)Gaussian massive spatial data with an application to Tukey-hh random fields estimation," Computational Statistics & Data Analysis, Elsevier, vol. 191(C).
    16. Brian J. Reich & Shu Yang & Yawen Guan & Andrew B. Giffin & Matthew J. Miller & Ana Rappold, 2021. "A Review of Spatial Causal Inference Methods for Environmental and Epidemiological Applications," International Statistical Review, International Statistical Institute, vol. 89(3), pages 605-634, December.
    17. Ephraim M. Hanks, 2017. "Modeling Spatial Covariance Using the Limiting Distribution of Spatio-Temporal Random Walks," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 112(518), pages 497-507, April.
    18. Matteo Tomasetto & Eleonora Arnone & Laura M. Sangalli, 2024. "Modeling Anisotropy and Non‐Stationarity Through Physics‐Informed Spatial Regression," Environmetrics, John Wiley & Sons, Ltd., vol. 35(8), December.
    19. Bondo, Kristin J. & Rosenberry, Christopher S. & Stainbrook, David & Walter, W. David, 2024. "Comparing risk of chronic wasting disease occurrence using Bayesian hierarchical spatial models and different surveillance types," Ecological Modelling, Elsevier, vol. 493(C).
    20. Scott, Ryan P., 2018. "Should we call the neighbors? Voluntary deliberation and citizen complaints about oil and gas drilling," Energy Policy, Elsevier, vol. 115(C), pages 258-272.

    More about this item

    Keywords

    ;
    ;
    ;

    Statistics

    Access and download statistics

    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:stapro:v:171:y:2021:i:c:s0167715221000031. 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/622892/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.