Local semicircle law with imprimitive variance matrix. (English) Zbl 1310.15069

Ajanki, Oskari Heikki; Erdős, Lászlo; Krüger, Torben

Electron. Commun. Probab. 19, Paper No. 33, 9 p. (2014).

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)

Keywords:

generalised Wigner matrices; generalised random sample covariance matrices; hard edge; local semicircle law