Functional CLT of eigenvectors for large sample covariance matrices

Abstract

In order to investigate property of the eigenvector matrix of sample covariance matrix $$\mathbf {S}_n$$ , in this paper, we establish the central limit theorem of linear spectral statistics associated with a new form of empirical spectral distribution $$H^{\mathbf {S}_n}$$ , based on eigenvectors and eigenvalues of sample covariance matrix $$\mathbf {S}_n$$ . Using Bernstein polynomial approximations, we prove the central limit theorem for linear spectral statistics of $$H^{\mathbf {S}_n}$$ , indexed by a set of functions with continuous third order derivatives over an interval including the support of Marcenko–Pastur law. This result provides further evidences to support the conjecture that the eigenmatrix of sample covariance matrix is asymptotically Haar distributed.