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The limiting spectral distribution of the product of the Wigner matrix and a nonnegative definite matrix

Author

Listed:
  • Bai, Z.D.
  • Zhang, L.X.

Abstract

Let Wn be n×n Hermitian whose entries on and above the diagonal are independent complex random variables satisfying the Lindeberg type condition. Let Tn be n×n nonnegative definitive and be independent of Wn. Assume that almost surely, as n-->[infinity], the empirical distribution of the eigenvalues of Tn converges weakly to a non-random probability distribution. Let . Then with the aid of the Stieltjes transforms, we show that almost surely, as n-->[infinity], the empirical distribution of the eigenvalues of An also converges weakly to a non-random probability distribution, a system of two equations determining the Stieltjes transform of the limiting distribution. Important analytic properties of this limiting spectral distribution are then derived by means of those equations. It is shown that the limiting spectral distribution is continuously differentiable everywhere on the real line except only at the origin and that a necessary and sufficient condition is available for determining its support. At the end, the density function of the limiting spectral distribution is calculated for two important cases of Tn, when Tn is a sample covariance matrix and when Tn is the inverse of a sample covariance matrix.

Suggested Citation

  • 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.
  • Handle: RePEc:eee:jmvana:v:101:y:2010:i:9:p:1927-1949
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    References listed on IDEAS

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    1. 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.
    2. Silverstein, J. W. & Choi, S. I., 1995. "Analysis of the Limiting Spectral Distribution of Large Dimensional Random Matrices," Journal of Multivariate Analysis, Elsevier, vol. 54(2), pages 295-309, August.
    3. 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.
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    Cited by:

    1. 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.
    2. Bao, Zhigang, 2012. "Strong convergence of ESD for the generalized sample covariance matrices when p/n→0," Statistics & Probability Letters, Elsevier, vol. 82(5), pages 894-901.

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