Central Limit Theorem for traces of the resolvents of half-heavy tailed Sample Covariance matrices

TL;DR

This paper establishes a Central Limit Theorem for the traces of resolvents of sample covariance matrices with half-heavy tailed entries, extending understanding of their spectral behavior without requiring finite fourth moments.

Contribution

It introduces a CLT for the Stieltjes transform of such matrices and their overlaps, providing new insights into their spectral fluctuations under heavy-tailed conditions.

Findings

CLT for the Stieltjes transform of half-heavy tailed sample covariance matrices

Explicit covariance kernel derived for the CLT

Extension of CLT to overlapping matrices

Abstract

We consider the spectrum of the Sample Covariance matrix $\mathbf{A}_N:= \frac{\mathbf{X}_N \mathbf{X}_N^*}{N}, $ where $\mathbf{X}_N$ is the $P\times N$ matrix with i.i.d. half-heavy tailed entries and $\frac{P}{N}\to y>0$ (the entries of the matrix have variance, but do not have the fourth moment). We derive the Central Limit Theorem for the Stieltjes transform of the matrix $\mathbf{A}_N$ and compute the covariance kernel. Apart from that, we derive the Central Limit Theorem for the Stieltjes transform of overlapping Sample Covariance matrices.