Tracy-Widom limit for the largest eigenvalue of a large class of complex Wishart matrices
Abstract
We study the limiting behavior of the largest eigenvalue of a large class of complex Wishart matrices. In other words, let X be an n*p matrix, and let its rows be i.i.d complex normal N_{C}(0,Sigma_p). We denote by H_p the spectral distribution of Sigma_p, and call lambda_i's its ordered eigenvalues. Let us call l_i's the ordered eigenvalues of X^*X and c the unique root in [0,1/lambda_1(Sigma_p)) of the equation \int ((lambda c)/(1-\lambda c))^2 dH_p(lambda) = n/p. The main result of this paper is that, under technical conditions on (Sigma_p,n,p), we have, when n->\infty, (l_1(X^*X)-n mu)/(n^{1/3} sigma) -> TW_2 . We give explicit formulas for mu and sigma, that depend non trivially on c. Here TW_2 denotes the Tracy-Widom law appearing in the study of the Gaussian Unitary Ensemble. This theorem applies to a number of covariance models found in applications, including well-behaved Toeplitz matrices and covariance matrices whose spectral distribution is a sum of atoms (under some conditions on the mass of the atoms). Generalizations of the theorem to certain spiked versions of models in G and a.s statements about l_1/n are given. Most known examples of convergence of the largest eigenvalue of a complex sample covariance matrix to this Tracy-Widom law are subcases of this result.Keywords
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