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Local semicircle law with imprimitive variance matrix. (English) Zbl 1310.15069
Summary: We extend the proof of the local semicircle law for generalized Wigner matrices given in [L. Erdős et al., Electron. J. Probab. 18, Paper No. 59, 58 p. (2013; Zbl 1373.15053)] 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 $$X^\ast X$$, where the variances of the entries of $$X$$ may vary.

##### MSC:
 15B52 Random matrices (algebraic aspects) 60B20 Random matrices (probabilistic aspects)
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