# 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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## References

SHOWING 1-4 OF 4 REFERENCES
On the empirical distribution of eigenvalues of a class of large dimensional random matrices
• Mathematics
• 1995
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),
Analysis of the limiting spectral distribution of large dimensional random matrices
• Mathematics
• 1995
Results on the analytic behavior of the limiting spectral distribution of matrices of sample covariance type, studied in Marcenko and Pastur [2] and Yin [8], are derived. Through an equation defining
DISTRIBUTION OF EIGENVALUES FOR SOME SETS OF RANDOM MATRICES
• Mathematics
• 1967
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