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On the empirical distribution of eigenvalues of large dimensional information-plus-noise-type matrices

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  • Dozier, R. Brent
  • Silverstein, Jack W.
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    Abstract

    Let Xn be nxN containing i.i.d. complex entries and unit variance (sum of variances of real and imaginary parts equals 1), [sigma]>0 constant, and Rn an nxN random matrix independent of Xn. Assume, almost surely, as n-->[infinity], the empirical distribution function (e.d.f.) of the eigenvalues of converges in distribution to a nonrandom probability distribution function (p.d.f.), and the ratio tends to a positive number. Then it is shown that, almost surely, the e.d.f. of the eigenvalues of converges in distribution. The limit is nonrandom and is characterized in terms of its Stieltjes transform, which satisfies a certain equation.

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    Bibliographic Info

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

    Volume (Year): 98 (2007)
    Issue (Month): 4 (April)
    Pages: 678-694

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    Handle: RePEc:eee:jmvana:v:98:y:2007:i:4:p:678-694

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    Keywords: Random matrix Empirical distribution function of eigenvalues Stieltjes transform;

    References

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    1. Silverstein, J. W. & Bai, Z. D., 1995. "On the Empirical Distribution of Eigenvalues of a Class of Large Dimensional Random Matrices," Journal of Multivariate Analysis, Elsevier, vol. 54(2), pages 175-192, August.
    2. Silverstein, J. W., 1995. "Strong Convergence of the Empirical Distribution of Eigenvalues of Large Dimensional Random Matrices," Journal of Multivariate Analysis, Elsevier, vol. 55(2), pages 331-339, November.
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    Cited by:
    1. Shabalin, Andrey A. & Nobel, Andrew B., 2013. "Reconstruction of a low-rank matrix in the presence of Gaussian noise," Journal of Multivariate Analysis, Elsevier, vol. 118(C), pages 67-76.

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