Random covariance matrices: universality of local statistics of eigenvalues (Q428150): Difference between revisions

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 \(n, p\). As 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.