Strong convergence of the empirical distribution of eigenvalues of large dimensional random matrices

@article{Silverstein1995StrongCO,
  title={Strong convergence of the empirical distribution of eigenvalues of large dimensional random matrices},
  author={Jack W. Silverstein},
  journal={Journal of Multivariate Analysis},
  year={1995},
  volume={55},
  pages={331-339}
}
  • J. W. Silverstein
  • Published 1 November 1995
  • Mathematics
  • Journal of Multivariate Analysis
Let X be n - N containing i.i.d. complex entries with E X11 - EX112 = 1, and T an n - n random Hermitian nonnegative definite, independent of X. Assume, almost surely, as n --> [infinity], the empirical distribution function (e.d.f.) of the eigenvalues of T converges in distribution, and the ratio n/N tends to a positive number. Then it is shown that, almost surely, the e.d.f. of the eigenvalues of (1/N) XX*T converges in distribution. The limit is nonrandom and is characterized in terms of its… 
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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),…
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In this paper we study the distribution of eigenvalues for two sets of random Hermitian matrices and one set of random unitary matrices. The statement of the problem as well as its method of…