Unbounded largest eigenvalue of large sample covariance matrices: Asymptotics, fluctuations and applications

Eigenvalue Distribution of Large Sample Covariance Matrices of Linear Processes

We derive the distribution of the eigenvalues of a large sample covariance matrix when the data is dependent in time. More precisely, the dependence for each variable i = 1, . . . , p is modelled as a linear process (Xi,t)t=1,...,n = ( ∑∞ j=0 cjZi,t−j)t=1,...,n, where {Zi,t} are assumed to be independent random variables with finite fourth moments. If the sample size n and the number of variabl...

Largest eigenvalues and 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...

Universality of Local Eigenvalue Statistics for Some Sample Covariance Matrices

Abstract We consider random, complex sample covariance matrices 1 N X ∗X , where X is a p×N random matrix with i.i.d. entries of distribution μ. It has been conjectured that both the distribution of the distance between nearest neighbor eigenvalues in the bulk and that of the smallest eigenvalues become, in the limit N → ∞, p N → 1, the same as that identified for a complex Gaussian distributio...

On Asymptotics of Eigenvectors of Large Sample Covariance Matrix

Let {Xij}, i, j = . . . , be a double array of i.i.d. complex random variables with EX11 = 0,E|X11| 2 = 1 and E|X11| 4 <∞, and let An = 1 N T 1/2 n XnX ∗ nT 1/2 n , where T 1/2 n is the square root of a nonnegative definite matrix Tn and Xn is the n×N matrix of the upper-left corner of the double array. The matrix An can be considered as a sample covariance matrix of an i.i.d. sample from a pop...