Gaussian Fluctuations for Sample Covariance Matrices with Dependent Data

TL;DR

This paper extends the central limit theorem for traces of powers of sample covariance matrices to cases where data rows and columns exhibit dependence, showing that Gaussian fluctuations persist under certain dependence structures.

Contribution

It demonstrates that the CLT for traces of powers of sample covariance matrices holds even with dependent data, broadening the applicability of spectral distribution results.

Findings

CLT for traces extends to dependent data

Gaussian fluctuations are preserved under dependence

Spectral distribution convergence remains valid

Abstract

It is known (Hofmann-Credner and Stolz (2008)) that the convergence of the mean empirical spectral distribution of a sample covariance matrix W_n = 1/n Y_n Y_n^t to the Mar\v{c}enko-Pastur law remains unaffected if the rows and columns of Y_n exhibit some dependence, where only the growth of the number of dependent entries, but not the joint distribution of dependent entries needs to be controlled. In this paper we show that the well-known CLT for traces of powers of W_n also extends to the dependent case.