IDEAS home Printed from https://ideas.repec.org/a/plo/pone00/0297150.html
   My bibliography  Save this article

A study on the spectral properties of covariance type matrices with isotropic log-concave column vectors

Author

Listed:
  • Shaojia Jin
  • Zhanwen Shi

Abstract

The limiting spectral distribution of matrix B n = 1 N X n X n * T n is considered in this paper. Existing results always focus on the condition of modifying Tn, but for Xn, it is usually assumed to be a matrix composed of n × N independent identically distributed elements. Here we specify the joint distribution of column vectors of Xn. In particular, entries on the same column of Xn are correlated, in contrast with more common independence assumptions. Assuming that the columns of Xn are random vectors following the isotropic log-concave distribution, and under some additional regularity conditions, we prove that the empirical spectral distribution F B n of matrix Bn converges to a deterministic probability distribution F almost surely. Moreover, the Stieltjes transformation m = m(z) of F satisfies a deterministic form of equation, and for any z ∈ C +, it is the unique solution of the equation.

Suggested Citation

  • Shaojia Jin & Zhanwen Shi, 2024. "A study on the spectral properties of covariance type matrices with isotropic log-concave column vectors," PLOS ONE, Public Library of Science, vol. 19(1), pages 1-11, January.
    • Handle: RePEc:plo:pone00:0297150
    DOI: 10.1371/journal.pone.0297150
    as

    Download full text from publisher

    File URL: https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0297150
    Download Restriction: no
    File URL: https://journals.plos.org/plosone/article/file?id=10.1371/journal.pone.0297150&type=printable
    Download Restriction: no
    File URL: https://libkey.io/10.1371/journal.pone.0297150?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
    ---><---

    References listed on IDEAS

    as
    1. Yin, Y. Q. & Krishnaiah, P. R., 1983. "A limit theorem for the eigenvalues of product of two random matrices," Journal of Multivariate Analysis, Elsevier, vol. 13(4), pages 489-507, December.
    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. Merlevède, F. & Peligrad, M., 2016. "On the empirical spectral distribution for matrices with long memory and independent rows," Stochastic Processes and their Applications, Elsevier, vol. 126(9), pages 2734-2760.
    2. Bai, Zhidong & Silverstein, Jack W., 2022. "A tribute to P.R. Krishnaiah," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    3. Bai, Z.D. & Miao, Baiqi & Jin, Baisuo, 2007. "On limit theorem for the eigenvalues of product of two random matrices," Journal of Multivariate Analysis, Elsevier, vol. 98(1), pages 76-101, January.
    4. Jin, Baisuo & Wang, Cheng & Miao, Baiqi & Lo Huang, Mong-Na, 2009. "Limiting spectral distribution of large-dimensional sample covariance matrices generated by VARMA," Journal of Multivariate Analysis, Elsevier, vol. 100(9), pages 2112-2125, October.
    5. Claudio Heinrich & Mark Podolskij, 2014. "On spectral distribution of high dimensional covariation matrices," CREATES Research Papers 2014-54, Department of Economics and Business Economics, Aarhus University.
    6. Olivier Ledoit & Sandrine P�ch�, 2009. "Eigenvectors of some large sample covariance matrices ensembles," IEW - Working Papers 407, Institute for Empirical Research in Economics - University of Zurich.
    7. M. Capitaine, 2013. "Additive/Multiplicative Free Subordination Property and Limiting Eigenvectors of Spiked Additive Deformations of Wigner Matrices and Spiked Sample Covariance Matrices," Journal of Theoretical Probability, Springer, vol. 26(3), pages 595-648, September.
    8. Fisher, Thomas J. & Sun, Xiaoqian & Gallagher, Colin M., 2010. "A new test for sphericity of the covariance matrix for high dimensional data," Journal of Multivariate Analysis, Elsevier, vol. 101(10), pages 2554-2570, November.
    9. Robert, Christian Y. & Rosenbaum, Mathieu, 2010. "On the limiting spectral distribution of the covariance matrices of time-lagged processes," Journal of Multivariate Analysis, Elsevier, vol. 101(10), pages 2434-2451, November.

    More about this item

    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:plo:pone00:0297150. 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: plosone (email available below). General contact details of provider: https://journals.plos.org/plosone/ .

    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.