Random covariance matrices: universality of local statistics of eigenvalues
type: Item
instance of: scholarly article
title: Random covariance matrices: universality of local statistics of eigenvalues
zbMATH Open document ID: 1247.15036
DOI: 10.1214/11-AOP648
published in: The Annals of Probability
publication date: 2012-06-19T00:00:00Z
full work available at URL: https://arxiv.org/abs/0912.0966
full work available at URL: https://projecteuclid.org/euclid.aop/1336136064
Mathematics Subject Classification ID: 15B52
Mathematics Subject Classification ID: 62J10
Mathematics Subject Classification ID: 60B20
zbMATH DE Number: 6047805
zbMATH Keywords: four moment theorem
zbMATH Keywords: universality
zbMATH Keywords: covariance matrices
zbMATH Keywords: asymptotic local eigenvalue statistics
zbMATH Keywords: large random matrices
zbMATH Keywords: Wigner matrices
zbMATH Keywords: random Hermitian matrix
author: Terence Tao
author: Van Vu
MaRDI profile type: Publication
arXiv ID: 0912.0966
cites work: Q3551026
cites work: Universality of local eigenvalue statistics for some sample covariance matrices
cites work: Eigenvalues and Condition Numbers of Random Matrices
cites work: Bulk universality for Wigner matrices
cites work: Bulk universality for Wigner Hermite matrices with subexponential decay
cites work: Universality of sine-kernel for Wigner matrices with a small Gaussian perturbation
cites work: Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices
cites work: Local semicircle law and complete delocalization for Wigner random matrices
cites work: Wegner Estimate and Level Repulsion for Wigner Random Matrices
cites work: The local relaxation flow approach to universality of the local statistics for random matrices
cites work: Bulk universality for generalized Wigner matrices
cites work: Rigidity of eigenvalues of generalized Wigner matrices
cites work: A universality result for the smallest eigenvalues of certain sample covariance matrices
cites work: The spectrum edge of random matrix ensembles.
cites work: Rate of convergence to the semi-circular law
cites work: Rate of convergence in probability to the Marchenko-Pastur law
cites work: Concentration of the spectral measure for large matrices
cites work: Gaussian fluctuations of eigenvalues in the GUE
cites work: Universality of the local spacing distribution in certain ensembles of Hermitian Wigner matrices
cites work: The concentration of measure phenomenon
cites work: Q5531480
cites work: SPECTRA OF RANDOM SELF ADJOINT OPERATORS
cites work: A note on universality of the distribution of the largest eigenvalues in certain sample covariance matrices
cites work: Random matrices: The distribution of the smallest singular values
cites work: Random matrices: Universality of local eigenvalue statistics up to the edge
cites work: Random matrices: universality of local eigenvalue statistics
cites work: The strong limits of random matrix spectra for sample matrices of independent elements
cites work: Limiting spectral distribution for a class of random matrices
OpenAlex ID: W3100430377
Recommended article: Random matrices: universality of local eigenvalue statistics
Recommended article: Bulk universality for deformed Wigner matrices
Recommended article: Universality of sine-kernel for Wigner matrices with a small Gaussian perturbation
Recommended article: A note on the central limit theorem for the eigenvalue counting function of Wigner matrices
Recommended article: Universality of local spectral statistics of random matrices
Recommended article: Q4887482
Recommended article: On universality of bulk local regime of the Hermitian sample covariance matrices
Recommended article: Moderate deviations for the eigenvalue counting function of Wigner matrices
Recommended article: Beyond universality in random matrix theory
Recommended article: Spectral properties of large random matrices with independent entries
review text: The asymptotic local eigenvalue statistics of covariance matrices of large random matrices are studied. The covariance matrices are given by the relation \(\frac{1}{n}M^{*}M\), where the rectangular matrix \(M=M_{n,p}=(\zeta_{ij})_{1 \leq i \leq p; 1 \leq j \leq n}\) has entries which are identically and independently distributed random variables with mean zero, variance one and have finite \(C_0\)th moment for some sufficiently large constant \(C_0\) independent of \(n, p\).NEWLINENEWLINEAs an analogue of the recent result of \textit{T. Tao} and \textit{V. Vu} [Acta Math.~206, No. 1, 127--204 (2011; Zbl 1217.15043)] for Wigner matrices a four moment theorem is established which asserts that the asymptotic behavior of local statistics of the eigenvalues of the covariance matrix are determined by the first four moments of the matrix entries. This theorem combined together with existing partial results for special ensembles is then used for establishing universality of local statistics of eigenvalues of covariance matrices. These results are also extended to the case where \(M\) is a random Hermitian matrix whose entries have finite \(C_0\)th moment without requiring the exponential decay of the atom distribution.
description: scientific article; zbMATH DE number 6047805