Local law for random Gram matrices

J. Alt, L. Erdös, T.H. Krüger, Electronic Journal of Probability 22 (2017) 25.

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Journal Article | Published | English
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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∗.
Publishing Year
Date Published
2017-03-08
Journal Title
Electronic Journal of Probability
Volume
22
Article Number
25
ISSN
IST-REx-ID

Cite this

Alt J, Erdös L, Krüger TH. Local law for random Gram matrices. Electronic Journal of Probability. 2017;22:25. doi:10.1214/17-EJP42
Alt, J., Erdös, L., & Krüger, T. H. (2017). Local law for random Gram matrices. Electronic Journal of Probability, 22, 25. https://doi.org/10.1214/17-EJP42
Alt, Johannes, László Erdös, and Torben H Krüger. “Local Law for Random Gram Matrices.” Electronic Journal of Probability 22 (2017): 25. https://doi.org/10.1214/17-EJP42.
J. Alt, L. Erdös, and T. H. Krüger, “Local law for random Gram matrices,” Electronic Journal of Probability, vol. 22, p. 25, 2017.
Alt J, Erdös L, Krüger TH. 2017. Local law for random Gram matrices. Electronic Journal of Probability. 22, 25.
Alt, Johannes, et al. “Local Law for Random Gram Matrices.” Electronic Journal of Probability, vol. 22, Institute of Mathematical Statistics, 2017, p. 25, doi:10.1214/17-EJP42.
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