Open Access
2017 Local law for random Gram matrices
Johannes Alt, László Erdős, Torben Krüger
Electron. J. Probab. 22: 1-41 (2017). DOI: 10.1214/17-EJP42


We prove a local law in the bulk of the spectrum for random Gram matrices $XX^*$, a generalization of sample covariance matrices, where $X$ is a large matrix with independent, centered entries with arbitrary variances. The limiting eigenvalue density that generalizes the Marchenko-Pastur law is determined by solving a system of nonlinear equations. Our entrywise and averaged local laws are on the optimal scale with the optimal error bounds. They hold both in the square case (hard edge) and in the properly rectangular case (soft edge). In the latter case we also establish a macroscopic gap away from zero in the spectrum of $XX^*$.


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Johannes Alt. László Erdős. Torben Krüger. "Local law for random Gram matrices." Electron. J. Probab. 22 1 - 41, 2017.


Received: 22 July 2016; Accepted: 27 February 2017; Published: 2017
First available in Project Euclid: 8 March 2017

zbMATH: 1376.60014
MathSciNet: MR3622895
Digital Object Identifier: 10.1214/17-EJP42

Primary: 15B52 , 60B20

Keywords: capacity of MIMO channels , general variance profile , hard edge , Marchenko-Pastur law , soft edge

Vol.22 • 2017
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