Local semicircle law with imprimitive variance matrix

Oskari Heikki Ajanki (Institute of Science and Technology Austria)
Lászlo Erdős (Institute of Science and Technology Austria)
Torben Krüger (Institute of Science and Technology Austria)

Abstract


We extend the proof of the local semicircle law for generalized Wigner matrices given in MR3068390 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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Pages: 1-9

Publication Date: June 9, 2014

DOI: 10.1214/ECP.v19-3121

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