Electronic Journal of Probability

Local law for random Gram matrices

Johannes Alt, László Erdős, and Torben Krüger

Full-text: Open access

Abstract

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 $XX^*$.

Article information

Source
Electron. J. Probab., Volume 22 (2017), paper no. 25, 41 pp.

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1488942016

Digital Object Identifier
doi:10.1214/17-EJP42

Mathematical Reviews number (MathSciNet)
MR3622895

Zentralblatt MATH identifier
1376.60014

Subjects
Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 15B52: Random matrices

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

Rights
Creative Commons Attribution 4.0 International License.

Citation

Alt, Johannes; Erdős, László; Krüger, Torben. Local law for random Gram matrices. Electron. J. Probab. 22 (2017), paper no. 25, 41 pp. doi:10.1214/17-EJP42. https://projecteuclid.org/euclid.ejp/1488942016


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