Large complex correlated Wishart matrices: Fluctuations and asymptotic independence at the edges

Abstract

We study the asymptotic behavior of eigenvalues of large complex correlated Wishart matrices at the edges of the limiting spectrum. In this setting, the support of the limiting eigenvalue distribution may have several connected components. Under mild conditions for the population matrices, we show that for every generic positive edge of that support, there exists an extremal eigenvalue which converges almost surely toward that edge and fluctuates according to the Tracy-Widom law at the scale $N^{2/3}$. Moreover, given several generic positive edges, we establish that the associated extremal eigenvalue fluctuations are asymptotically independent. Finally, when the leftmost edge is the origin (hard edge), the fluctuations of the smallest eigenvalue are described by mean of the Bessel kernel at the scale $N^2$.

Published at http://dx.doi.org/10.1214/15-AOP1022 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (5)

• Central limit theorems for eigenvalues in a spiked population model
• The Tracy--Widom limit for the largest eigenvalues of singular complex Wishart matrices
• Gaussian fluctuations for linear spectral statistics of large random covariance matrices
• Large Complex Correlated Wishart Matrices: The Pearcey Kernel and Expansion at the Hard Edge
• On the principal components of sample covariance matrices

Cited by in corpus (9)

• Deterministic parallel analysis: An improved method for selecting factors and principal components
• A necessary and sufficient condition for edge universality at the largest singular values of covariance matrices
• Large Complex Correlated Wishart Matrices: The Pearcey Kernel and Expansion at the Hard Edge
• The shape of shortest paths in random spatial networks
• Convergence of eigenvector empirical spectral distribution of sample covariance matrices
• Asymptotics of Fredholm determinant associated with the Pearcey kernel
• Local circular law for the product of a deterministic matrix with a random matrix
• Cusp Universality for Random Matrices I: Local Law and the Complex Hermitian Case
• Eigenvector overlaps in large sample covariance matrices and nonlinear shrinkage estimators