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.
Citation
Oskari Ajanki. Lászlo Erdős. Torben Krüger. "Local semicircle law with imprimitive variance matrix." Electron. Commun. Probab. 19 1 - 9, 2014. https://doi.org/10.1214/ECP.v19-3121
Information