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September 2005 Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices
Jinho Baik, Gérard Ben Arous, Sandrine Péché
Ann. Probab. 33(5): 1643-1697 (September 2005). DOI: 10.1214/009117905000000233

Abstract

We compute the limiting distributions of the largest eigenvalue of a complex Gaussian sample covariance matrix when both the number of samples and the number of variables in each sample become large. When all but finitely many, say r, eigenvalues of the covariance matrix are the same, the dependence of the limiting distribution of the largest eigenvalue of the sample covariance matrix on those distinguished r eigenvalues of the covariance matrix is completely characterized in terms of an infinite sequence of new distribution functions that generalize the Tracy–Widom distributions of the random matrix theory. Especially a phase transition phenomenon is observed. Our results also apply to a last passage percolation model and a queueing model.

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Jinho Baik. Gérard Ben Arous. Sandrine Péché. "Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices." Ann. Probab. 33 (5) 1643 - 1697, September 2005. https://doi.org/10.1214/009117905000000233

Information

Published: September 2005
First available in Project Euclid: 22 September 2005

zbMATH: 1086.15022
MathSciNet: MR2165575
Digital Object Identifier: 10.1214/009117905000000233

Subjects:
Primary: 15A52, 41A60, 60F99, 62E20, 62H20

Rights: Copyright © 2005 Institute of Mathematical Statistics

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Vol.33 • No. 5 • September 2005
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