IDEAS home Printed from https://ideas.repec.org/a/spr/jotpro/v25y2012i4d10.1007_s10959-010-0332-5.html
   My bibliography  Save this article

Schoenberg’s Theorem and Unitarily Invariant Random Arrays

Author

Listed:
  • Olav Kallenberg

    (Auburn University)

Abstract

Motivated by Schoenberg’s theorem in classical analysis and some recent work on the circular law, we extend the basic representations of unitarily invariant random arrays and functionals to the complex case. As in the real case, the basic building blocks are multiple Wiener–Itô integrals on tensor products of a separable Hilbert space. The complex setting leads to some unexpected simplifications.

Suggested Citation

  • Olav Kallenberg, 2012. "Schoenberg’s Theorem and Unitarily Invariant Random Arrays," Journal of Theoretical Probability, Springer, vol. 25(4), pages 1013-1039, December.
    • Handle: RePEc:spr:jotpro:v:25:y:2012:i:4:d:10.1007_s10959-010-0332-5
    DOI: 10.1007/s10959-010-0332-5
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10959-010-0332-5
    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/s10959-010-0332-5?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. Aldous, David J., 1981. "Representations for partially exchangeable arrays of random variables," Journal of Multivariate Analysis, Elsevier, vol. 11(4), pages 581-598, December.
    2. Dawid, A. P., 1978. "Extendibility of spherical matrix distributions," Journal of Multivariate Analysis, Elsevier, vol. 8(4), pages 559-566, December.
    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. Yanik-Pascal Förster & Mario Kieburg & Holger Kösters, 2021. "Polynomial Ensembles and Pólya Frequency Functions," Journal of Theoretical Probability, Springer, vol. 34(4), pages 1917-1950, December.

    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. repec:hal:journl:hal-04672521 is not listed on IDEAS
    2. Bikramjit Das & Tiandong Wang & Gengling Dai, 2022. "Asymptotic Behavior of Common Connections in Sparse Random Networks," Methodology and Computing in Applied Probability, Springer, vol. 24(3), pages 2071-2092, September.
    3. Olav Kallenberg, 1999. "Multivariate Sampling and the Estimation Problem for Exchangeable Arrays," Journal of Theoretical Probability, Springer, vol. 12(3), pages 859-883, July.
    4. Dmitry Arkhangelsky & Guido Imbens, 2023. "Causal Models for Longitudinal and Panel Data: A Survey," Papers 2311.15458, arXiv.org, revised Jun 2024.
    5. Bryan S. Graham, 2020. "Sparse network asymptotics for logistic regression," Papers 2010.04703, arXiv.org.
    6. Bryan S. Graham, 2019. "Network Data," Papers 1912.06346, arXiv.org.
    7. François Caron & Emily B. Fox, 2017. "Sparse graphs using exchangeable random measures," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(5), pages 1295-1366, November.
    8. Kameswarrao S. Casukhela, 1997. "Symmetric Distributions of Random Measures in Higher Dimensions," Journal of Theoretical Probability, Springer, vol. 10(3), pages 759-771, July.
    9. Bryan S. Graham & Fengshi Niu & James L. Powell, 2019. "Kernel density estimation for undirected dyadic data," CeMMAP working papers CWP39/19, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    10. Bryan S. Graham, 2019. "Dyadic Regression," Papers 1908.09029, arXiv.org.
    11. Graham, Bryan S. & Niu, Fengshi & Powell, James L., 2024. "Kernel density estimation for undirected dyadic data," Journal of Econometrics, Elsevier, vol. 240(2).
    12. Peter D. Hoff, 2009. "Multiplicative latent factor models for description and prediction of social networks," Computational and Mathematical Organization Theory, Springer, vol. 15(4), pages 261-272, December.
    13. Paolo Leonetti, 2018. "Finite Partially Exchangeable Laws Are Signed Mixtures of Product Laws," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 80(2), pages 195-214, August.
    14. Patrick Rubin‐Delanchy & Joshua Cape & Minh Tang & Carey E. Priebe, 2022. "A statistical interpretation of spectral embedding: The generalised random dot product graph," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 84(4), pages 1446-1473, September.
    15. Konstantin Tikhomirov & Pierre Youssef, 2019. "On the norm of a random jointly exchangeable matrix," Journal of Theoretical Probability, Springer, vol. 32(4), pages 1990-2005, December.
    16. Robert Lunde & Purnamrita Sarkar, 2023. "Subsampling sparse graphons under minimal assumptions," Biometrika, Biometrika Trust, vol. 110(1), pages 15-32.
    17. Yin, Mei, 2022. "Remarks on power-law random graphs," Stochastic Processes and their Applications, Elsevier, vol. 153(C), pages 183-197.
    18. Pe[combining cedilla]ski, Marcin, 2011. "Prior symmetry, similarity-based reasoning, and endogenous categorization," Journal of Economic Theory, Elsevier, vol. 146(1), pages 111-140, January.
    19. Kai-Tai Fang & Run-Ze Li, 1997. "Some methods for generating both an NT-net and the uniform distribution on a Stiefel manifold and their applications," Computational Statistics & Data Analysis, Elsevier, vol. 24(1), pages 29-46, March.
    20. Harold D Chiang & Yukun Ma & Joel Rodrigue & Yuya Sasaki, 2021. "Dyadic double/debiased machine learning for analyzing determinants of free trade agreements," Papers 2110.04365, arXiv.org, revised Dec 2022.
    21. Hsieh, Chih-Sheng & Hsu, Yu-Chin & Ko, Stanley I.M. & Kovářík, Jaromír & Logan, Trevon D., 2024. "Non-representative sampled networks: Estimation of network structural properties by weighting," Journal of Econometrics, Elsevier, vol. 240(1).

    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:25:y:2012:i:4:d:10.1007_s10959-010-0332-5. 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.