Limit distributions of sample covariance matrices are compound free Poisson

TL;DR

This paper demonstrates that the eigenvalue distribution of certain sample covariance matrices converges to a compound free Poisson distribution, extending previous results to more general random vectors.

Contribution

It establishes the weak convergence of eigenvalue distributions for sample covariance matrices of dependent random vectors with bounded fourth moments to a compound free Poisson law.

Findings

Eigenvalue distributions converge to a compound free Poisson distribution.

Results apply to dependent random vectors with bounded $L^4$ norms.

Extends classical results to more general dependence structures.

Abstract

We show that the empirical distribution of the eigenvalues of the sample covariance matrix of certain random vectors (not necessarily independent entries) with bounded marginal $L^{4}$ norms converges weakly to a compound free Poisson distribution.