Local law for random Gram matrices Journal Article


Author(s): Alt, Johannes; Erdős, László; Krüger, Torben
Article Title: Local law for random Gram matrices
Affiliation IST Austria
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∗.
Keywords: Hard edge; capacity of MIMO channels; Marchenko-Pastur law; soft edge; general variance profile
Journal Title: Electronic Journal of Probability
Volume: 22
ISSN: 1083-6489
Publisher: Institute of Mathematical Statistics  
Date Published: 2017-03-08
Start Page: Article Number: 25
Copyright Statement: CC BY 4.0
URL:
DOI: 10.1214/17-EJP42
Open access: yes (OA journal)
IST Austria Authors
  1. László Erdős
    102 Erdős
  2. Johannes Alt
    2 Alt
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