Eigenvectors of Sample Covariance Matrices: Universality of global fluctuations

Abstract

In this paper, we prove a universality result of convergence for a bivariate random process defined by the eigenvectors of a sample covariance matrix. Let $V_n=(v_{ij}{)}_{i\le n,j\le m}$ be a $n\times m$ random matrix, where $(n/m)\to y>0$ as $n\to \infty$, and let $X_n=(1/m)V_n{V}_n^{\ast }$ be the sample covariance matrix associated to $V_n$. Consider the spectral decomposition of $X_n$ given by $U_n{D}_n{U}_n^{\ast }$, where $U_n=(u_{ij}{)}_{n\times n}$ is an eigenmatrix of $X_n$. We prove, under some moments conditions, that the bivariate random process $<B_{s,t}^n=\underset{1\le j\le \lfloor nt\rfloor }{{\sum }_{1\le i\le \lfloor ns\rfloor }}<|u_{i,j}{|}^2-\frac{1}{n}>{>}_{(s,t)\in [0,1{]}^2}$ converges in distribution to a bivariate Brownian bridge. This type of result has been already proved for Wishart matrices (LOE/LUE) and Wigner matrices. This supports the intuition that the eigenmatrix of a sample covariance matrix is in a way "asymptotically Haar distributed". Our analysis follows closely the one of Benaych-Georges for Wigner matrices, itself inspired by Silverstein works on the eigenvectors of sample covariance matrices.

21 pages. arXiv admin note: substantial text overlap with arXiv:1104.1219 by other authors

References in corpus (2)

• Local semicircle law and complete delocalization for Wigner random matrices
• Random weighted projections, random quadratic forms and random eigenvectors

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