Electronic Journal of Probability
- Electron. J. Probab.
- Volume 22 (2017), paper no. 25, 41 pp.
Local law for random Gram matrices
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 $XX^*$.
Electron. J. Probab., Volume 22 (2017), paper no. 25, 41 pp.
Received: 22 July 2016
Accepted: 27 February 2017
First available in Project Euclid: 8 March 2017
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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