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On the Empirical Distribution of Eigenvalues of a Class of Large Dimensional Random Matrices

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  • Silverstein, J. W.
  • Bai, Z. D.
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    Abstract

    A stronger result on the limiting distribution of the eigenvalues of random Hermitian matrices of the form A + XTX*, originally studied in Marcenko and Pastur, is presented. Here, X(N - n), T(n - n), and A(N - N) are independent, with X containing i.i.d. entries having finite second moments, T is diagonal with real (diagonal) entries, A is Hermitian, and n/N --> c > 0 as N --> [infinity]. Under additional assumptions on the eigenvalues of A and T, almost sure convergence of the empirical distribution function of the eigenvalues of A + XTX* is proven with the aid of Stieltjes transforms, taking a more direct approach than previous methods.

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    Bibliographic Info

    Article provided by Elsevier in its journal Journal of Multivariate Analysis.

    Volume (Year): 54 (1995)
    Issue (Month): 2 (August)
    Pages: 175-192

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    Handle: RePEc:eee:jmvana:v:54:y:1995:i:2:p:175-192

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    Cited by:
    1. Guhlich, Matthias & Nagel, Jan & Dette, Holger, 2010. "Random block matrices generalizing the classical Jacobi and Laguerre ensembles," Journal of Multivariate Analysis, Elsevier, vol. 101(8), pages 1884-1897, September.
    2. Jean-Philippe Bouchaud & Laurent Laloux & M. Augusta Miceli & Marc Potters, 2005. "Large dimension forecasting models and random singular value spectra," Papers physics/0512090, arXiv.org.
    3. Rubio, Francisco & Mestre, Xavier, 2011. "Spectral convergence for a general class of random matrices," Statistics & Probability Letters, Elsevier, vol. 81(5), pages 592-602, May.
    4. Wang, Lili & Paul, Debashis, 2014. "Limiting spectral distribution of renormalized separable sample covariance matrices when p/n→0," Journal of Multivariate Analysis, Elsevier, vol. 126(C), pages 25-52.
    5. Paul, Debashis & Silverstein, Jack W., 2009. "No eigenvalues outside the support of the limiting empirical spectral distribution of a separable covariance matrix," Journal of Multivariate Analysis, Elsevier, vol. 100(1), pages 37-57, January.
    6. Olivier Ledoit & Sandrine Péché, 2009. "Eigenvectors of some large sample covariance matrices ensembles," IEW - Working Papers 407, Institute for Empirical Research in Economics - University of Zurich.
    7. Taras Bodnar & Nestor Parolya & Wolfgang Schmid, 2014. "Estimation of the Global Minimum Variance Portfolio in High Dimensions," Papers 1406.0437, arXiv.org.
    8. Bai, Z.D. & Zhang, L.X., 2010. "The limiting spectral distribution of the product of the Wigner matrix and a nonnegative definite matrix," Journal of Multivariate Analysis, Elsevier, vol. 101(9), pages 1927-1949, October.
    9. Wang, Cheng & Yang, Jing & Miao, Baiqi & Cao, Longbing, 2013. "Identity tests for high dimensional data using RMT," Journal of Multivariate Analysis, Elsevier, vol. 118(C), pages 128-137.
    10. Olivier Ledoit & Michael Wolf, 2013. "Spectrum estimation: a unified framework for covariance matrix estimation and PCA in large dimensions," ECON - Working Papers 105, Department of Economics - University of Zurich, revised Jul 2013.
    11. Li, Weiming & Qin, Yingli, 2014. "Hypothesis testing for high-dimensional covariance matrices," Journal of Multivariate Analysis, Elsevier, vol. 128(C), pages 108-119.
    12. Hsu, Chih-Yuan & Wu, Tiee-Jian, 2013. "Efficient estimation of the mode of continuous multivariate data," Computational Statistics & Data Analysis, Elsevier, vol. 63(C), pages 148-159.
    13. 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.
    14. Pan, Guangming & Miao, Boqi & Jin, Baisuo, 2005. "Some limiting theorems of some random quadratic forms," Statistics & Probability Letters, Elsevier, vol. 75(3), pages 151-157, December.
    15. Dozier, R. Brent & Silverstein, Jack W., 2007. "On the empirical distribution of eigenvalues of large dimensional information-plus-noise-type matrices," Journal of Multivariate Analysis, Elsevier, vol. 98(4), pages 678-694, April.
    16. Olivier Ledoit & Michael Wolf, 2013. "Optimal estimation of a large-dimensional covariance matrix under Stein’s loss," ECON - Working Papers 122, Department of Economics - University of Zurich, revised Dec 2013.

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