Functional CLT for sample covariance matrices

TL;DR

This paper proves a central limit theorem for linear spectral statistics of sample covariance matrices using Bernstein polynomial approximations, providing explicit formulas for asymptotic mean and covariance functions.

Contribution

It introduces a novel proof technique employing Bernstein polynomial approximations for CLT in spectral analysis of covariance matrices.

Findings

Established CLT for spectral statistics with functions having continuous fourth derivatives

Derived explicit formulas for asymptotic mean and covariance functions

Extended the CLT to functions supported on the Marčenko–Pastur law interval

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 Mar\u{c}enko--Pastur law. We also derive the explicit expressions for asymptotic mean and covariance functions.