Local semicircle law with imprimitive variance matrix

TL;DR

This paper extends the local semicircle law to matrices with an eigenvalue -1 in their variance matrix, enabling a simplified proof of the local Marchenko-Pastur law at the hard edge for sample covariance matrices with varying entry variances.

Contribution

It generalizes the local semicircle law to imprimitive variance matrices, providing a concise proof of the local Marchenko-Pastur law at zero for sample covariance matrices.

Findings

Extended the local semicircle law to matrices with eigenvalue -1 in variance matrix

Provided a short proof of the local Marchenko-Pastur law at zero for covariance matrices

Applicable to matrices with non-uniform variances of entries

Abstract

We extend the proof of the local semicircle law for generalized Wigner matrices given in [4] to the case when the matrix of variances has an eigenvalue $ -1 $. In particular, this result provides a short proof of the optimal local Marchenko-Pastur law at the hard edge (i.e. around zero) for sample covariance matrices $ \boldsymbol{\mathrm{X}}^\ast \boldsymbol{\mathrm{X}} $, where the variances of the entries of $ \boldsymbol{\mathrm{X}} $ may vary.