IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v185y2025ics0304414925000547.html

Intersections of Poisson k-flats in hyperbolic space: Completing the picture

Author

Listed:
  • Bühler, Tillmann
  • Hug, Daniel

Abstract

Let η be an isometry invariant Poisson process of k-flats, 0≤k≤d−1, in d-dimensional hyperbolic space. For d−m(d−k)≥0, the m-th order intersection process of η consists of all nonempty intersections of distinct flats E1,…,Em∈η. Of particular interest is the total volume Fr(m) of this intersection process in a ball of radius r. For 2k>d+1, we determine the asymptotic distribution of Fr(m), as r→∞, previously known only for m=1, and derive rates of convergence in the Kolmogorov distance. Properties of the non-Gaussian limit distribution are discussed. We further study the asymptotic covariance matrix of the vector (Fr(1),…,Fr(m))⊤.

Suggested Citation

  • Bühler, Tillmann & Hug, Daniel, 2025. "Intersections of Poisson k-flats in hyperbolic space: Completing the picture," Stochastic Processes and their Applications, Elsevier, vol. 185(C).
    • Handle: RePEc:eee:spapps:v:185:y:2025:i:c:s0304414925000547
    DOI: 10.1016/j.spa.2025.104613
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414925000547
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2025.104613?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. Bonnet, Gilles & Hirsch, Christian & Rosen, Daniel & Willhalm, Daniel, 2023. "Limit theory of sparse random geometric graphs in high dimensions," Stochastic Processes and their Applications, Elsevier, vol. 163(C), pages 203-236.
    2. Johan H. Tykesson, 2009. "Continuum Percolation at and above the Uniqueness Threshold on Homogeneous Spaces," Journal of Theoretical Probability, Springer, vol. 22(2), pages 402-417, June.
    3. Grygierek, Jens & Thäle, Christoph, 2020. "Gaussian fluctuations for edge counts in high-dimensional random geometric graphs," Statistics & Probability Letters, Elsevier, vol. 158(C).
    4. Betken, Carina & Hug, Daniel & Thäle, Christoph, 2023. "Intersections of Poisson k-flats in constant curvature spaces," Stochastic Processes and their Applications, Elsevier, vol. 165(C), pages 96-129.
    5. Gilles Bonnet & Eliza O'Reilly, 2022. "Facets of spherical random polytopes," Mathematische Nachrichten, Wiley Blackwell, vol. 295(10), pages 1901-1933, October.
    6. Molchanov,Ilya & Molinari,Francesca, 2018. "Random Sets in Econometrics," Cambridge Books, Cambridge University Press, number 9781107121201.
    7. Kabluchko, Zakhar & Marynych, Alexander & Raschel, Kilian, 2024. "Random walks in the high-dimensional limit II: The crinkled subordinator," Stochastic Processes and their Applications, Elsevier, vol. 176(C).
    8. Gusakova, Anna & Heiny, Johannes & Thäle, Christoph, 2023. "The volume of random simplices from elliptical distributions in high dimension," Stochastic Processes and their Applications, Elsevier, vol. 164(C), pages 357-382.
    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. Raffaella Giacomini & Toru Kitagawa, 2021. "Robust Bayesian Inference for Set‐Identified Models," Econometrica, Econometric Society, vol. 89(4), pages 1519-1556, July.
    2. Semenova, Vira, 2023. "Debiased machine learning of set-identified linear models," Journal of Econometrics, Elsevier, vol. 235(2), pages 1725-1746.
    3. Antonio Avilés López & José Miguel Zapata García, 2020. "Boolean Valued Representation of Random Sets and Markov Kernels with Application to Large Deviations," Mathematics, MDPI, vol. 8(10), pages 1-23, October.
    4. Merlo, Luca & Petrella, Lea & Salvati, Nicola & Tzavidis, Nikos, 2022. "Marginal M-quantile regression for multivariate dependent data," Computational Statistics & Data Analysis, Elsevier, vol. 173(C).
    5. Vira Semenova, 2023. "Debiased Machine Learning of Aggregated Intersection Bounds and Other Causal Parameters," Papers 2303.00982, arXiv.org, revised May 2025.
    6. Hiroaki Kaido & Yi Zhang, 2019. "Robust likelihood ratio tests for incomplete economic models," CeMMAP working papers CWP68/19, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    7. Martin Dumav & Maxwell B. Stinchcombe, 2021. "The multiple priors of the open-minded decision maker," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 71(2), pages 663-692, March.
    8. Diaye, Marc-Arthur & Koshevoy, Gleb A. & Molchanov, Ilya, 2019. "Lift expectations of random sets," Statistics & Probability Letters, Elsevier, vol. 145(C), pages 110-117.
    9. Levon Barseghyan & Maura Coughlin & Francesca Molinari & Joshua C. Teitelbaum, 2019. "Heterogeneous Choice Sets and Preferences," Papers 1907.02337, arXiv.org, revised Feb 2021.
    10. Bonnet, Gilles & Hirsch, Christian & Rosen, Daniel & Willhalm, Daniel, 2023. "Limit theory of sparse random geometric graphs in high dimensions," Stochastic Processes and their Applications, Elsevier, vol. 163(C), pages 203-236.
    11. S. Settepanella & A. Terni & M. Franciosi & L. Li, 2022. "The robustness of the generalized Gini index," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 45(2), pages 521-539, December.
    12. Laura Doval & Ran Eilat & Tianhao Liu & Yangfan Zhou, 2024. "Revealed Information," Papers 2411.13293, arXiv.org, revised Jun 2025.
    13. Daisuke Kurisu & Yuta Okamoto & Taisuke Otsu, 2025. "Difference-in-Differences with Interval Data," Papers 2512.08759, arXiv.org.
    14. Marc-Arthur Diaye & Gleb Koshevoy & Ilya Molchanov, 2019. "Lift expectations of random sets [Augmenter les attentes concernant les ensembles aléatoires]," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-03897964, HAL.
    15. Antonio Elías & Stanislav Nagy, 2025. "Statistical properties of partially observed integrated functional depths," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 34(1), pages 125-150, March.
    16. Undral Byambadalai, 2022. "Identification and Inference for Welfare Gains without Unconfoundedness," Papers 2207.04314, arXiv.org.
    17. Chesher, Andrew & Kim, Dongwoo & Rosen, Adam M., 2023. "IV methods for Tobit models," Journal of Econometrics, Elsevier, vol. 235(2), pages 1700-1724.
    18. Ilya Molchanov & Anja Muhlemann, 2019. "Nonlinear expectations of random sets," Papers 1903.04901, arXiv.org.
    19. Levon Barseghyan & Maura Coughlin & Francesca Molinari & Joshua C. Teitelbaum, 2021. "Heterogeneous Choice Sets and Preferences," Econometrica, Econometric Society, vol. 89(5), pages 2015-2048, September.
    20. Colubi, Ana & Ramos-Guajardo, Ana Belén, 2023. "Fuzzy sets and (fuzzy) random sets in Econometrics and Statistics," Econometrics and Statistics, Elsevier, vol. 26(C), pages 84-98.

    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:spapps:v:185:y:2025:i:c:s0304414925000547. 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.