Translator Disclaimer
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

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 $XX^*$.

Citation

Download Citation

Johannes Alt. László Erdős. Torben Krüger. "Local law for random Gram matrices." Electron. J. Probab. 22 1 - 41, 2017. https://doi.org/10.1214/17-EJP42

Information

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

Subjects:
Primary: 15B52, 60B20

JOURNAL ARTICLE
41 PAGES


SHARE
Vol.22 • 2017
Back to Top