Gaussian Fluctuations in Complex Sample Covariance Matrices

Gaussian fluctuations in complex sample covariance matrices

Let X = (Xi,j)m×n,m ≥ n, be a complex Gaussian random matrix with mean zero and variance 1 n , let S = XX be a sample covariance matrix. In this paper we are mainly interested in the limiting behavior of eigenvalues when m n → γ ≥ 1 as n → ∞. Under certain conditions on k, we prove the central limit theorem holds true for the k-th largest eigenvalues λ(k) as k tends to infinity as n → ∞. The pr...

A Clt for Regularized Sample Covariance Matrices

We consider the spectral properties of a class of regularized estimators of (large) empirical covariance matrices corresponding to stationary (but not necessarily Gaussian) sequences, obtained by banding. We prove a law of large numbers (similar to that proved in the Gaussian case by Bickel and Levina), which implies that the spectrum of a banded empirical covariance matrix is an efficient esti...

Gaussian fluctuations for linear spectral statistics of large random covariance matrices

Consider a N ×n matrix Σn = 1 √ n R 1/2 n Xn, where Rn is a nonnegative definite Hermitian matrix and Xn is a random matrix with i.i.d. real or complex standardized entries. The fluctuations of the linear statistics of the eigenvalues: Trace f(ΣnΣ ∗ n) = N ∑ i=1 f(λi), (λi) eigenvalues of ΣnΣ ∗ n, are shown to be gaussian, in the regime where both dimensions of matrix Σn go to infinity at the s...

Largest eigenvalues and sample covariance matrices

Spectral Density of Sparse Sample Covariance Matrices

Applying the replica method of statistical mechanics, we evaluate the eigenvalue density of the large random matrix (sample covariance matrix) of the form J = ATA, where A is an M × N real sparse random matrix. The difference from a dense random matrix is the most significant in the tail region of the spectrum. We compare the results of several approximation schemes, focusing on the behavior in...