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arXiv · 1311.2016

Local semicircle law with imprimitive variance matrix

Oskari Ajanki·Laszlo Erdos·Torben Krüger
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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.

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Oskari Ajanki, Laszlo Erdos, Torben Krüger. 2013-11-08. Local semicircle law with imprimitive variance matrix. https://arxiv.org/abs/1311.2016

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