# Analysis of the limiting spectral distribution of large dimensional random matrices

@article{Silverstein1995AnalysisOT,
title={Analysis of the limiting spectral distribution of large dimensional random matrices},
author={Jack W. Silverstein and Sang Il Choi},
journal={Journal of Multivariate Analysis},
year={1995},
volume={54},
pages={295-309}
}
• Published 1 August 1995
• Mathematics
• Journal of Multivariate Analysis
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 its Stieltjes transform, it is shown that the limiting distribution has a continuous derivative away from zero, the derivative being analytic wherever it is positive, and resembles [formula] for most cases of x0 in the boundary of its support. A complete analysis of a way to determine its support…
283 Citations
Limiting Spectral Distributions of Large Dimensional Random Matrices
• Mathematics
• 2005
Models where the number of parameters increases with the sample size, are becoming increasingly important in statistics. This necessitates a close look at the statistical properties of eigenvalues of
Strong convergence of the empirical distribution of eigenvalues of large dimensional random matrices
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
GAUSSIAN FLUCTUATIONS FOR LINEAR SPECTRAL STATISTICS OF LARGE RANDOM COVARIANCE MATRICES By
• Mathematics, Computer Science
• 2016
The main improvements with respect to Bai and Silverstein’s CLT are twofold: First, general entries with finite fourth moment are considered, but whose fourth cumulant is nonnull, that is, whose fourth moment may differ from the moment of a (real or complex) Gaussian random variable.
Eigenvectors of Some Large Sample Covariance Matrices Ensembles
• Mathematics
• 2009
We consider sample covariance matrices constructed from real or complex i.i.d. variates with finite 12th moment. We assume that the population covariance matrix is positive definite and its spectral
A LOCAL MOMENT ESTIMATOR OF THE SPECTRUM OF A LARGE DIMENSIONAL COVARIANCE MATRIX
• Mathematics
• 2013
This paper considers the problem of estimating the population spectral distribution from a sample covariance matrix in large dimensional situations. We generalize the contour-integral based method in
Analysis of the limiting spectral measure of large random matrices of the separable covariance type
• Mathematics
• 2013
Consider the random matrix $\Sigma = D^{1/2} X \widetilde D^{1/2}$ where D and $\widetilde D$ are deterministic Hermitian nonnegative matrices with respective dimensions N × N and n × n, and where X
Tracy–Widom limit for the largest eigenvalue of a large class of complex sample covariance matrices
We consider the asymptotic fluctuation behavior of the largest eigenvalue of certain sample covariance matrices in the asymptotic regime where both dimensions of the corresponding data matrix go to
A Survey on the Eigenvalues Local Behavior of Large Complex Correlated Wishart Matrices
• Mathematics
• 2015
The aim of this note is to provide a pedagogical survey of the recent works by the authors ( arXiv:1409.7548 and arXiv:1507.06013) concerning the local behavior of the eigenvalues of large complex
Tracy-Widom limit for the largest eigenvalue of a large class of complex Wishart matrices
The problem of understanding the limiting behavior of the largest eigenvalue of sample covariance matrices computed from data matrices for which both dimensions are large has recently attracted a lot

## References

SHOWING 1-8 OF 8 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),
The Strong Limits of Random Matrix Spectra for Sample Matrices of Independent Elements
This paper proves almost-sure convergence of the empirical measure of the normalized singular values of increasing rectangular submatrices of an infinite random matrix of independent elements. The
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
Signal detection via spectral theory of large dimensional random matrices
• Mathematics
IEEE Trans. Signal Process.
• 1992
The theoretical analysis presented focuses on the splitting of the spectrum of sample covariance matrix into noise and signal eigenvalues and it is shown that when the number of sensors is large thenumber of signals can be estimated with a sample size considerably less than that required by previous approaches.
The Limiting Eigenvalue Distribution of a Multivariate F Matrix
Let $X_{ij} ,Y_{ij} i,j = 1,2, \cdots$, be i.i.d. $N(0,1)$ random variables and for positive integers $p,m,n$, let $\bar X_p = (X_{ij} ) i = 1,2, \cdots ,p; j = 1,2, \cdots ,m$, and \$\bar Y_p =
Spectral theory of large dimensional random matrices applied to signal detection
• Technical Report,
• 1990