# Eigenvectors of Sample Covariance Matrices: Universality of global
fluctuations

Research paper by **Ali Bouferroum**

Indexed on: **18 Jun '13**Published on: **18 Jun '13**Published in: **Mathematics - Probability**

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#### 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
$V_n=(v_{ij})_{i \leq n,\, j\leq m}$ 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 $ be the
sample covariance matrix associated to $V_n \:$. Consider the spectral
decomposition of $X_n$ given by $ U_n D_n U_n^{*}$, 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\leq j \leq \lfloor nt \rfloor}{\sum_{1\leq i \leq \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.