Functional CLT for sample covariance matrices

Abstract

Using Bernstein polynomial approximations, we prove the central limit theorem for linear spectral statistics of sample covariance matrices, indexed by a set of functions with continuous fourth order derivatives on an open interval including $[(1-\sqrt{y})^{2},(1+\sqrt{y})^{2}]$, the support of the Marčenko–Pastur law. We also derive the explicit expressions for asymptotic mean and covariance functions.