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.
%0 Journal Article
%1 baik2005phase
%A Baik, Jinho
%A Ben Arous, Gérard
%A Péché, Sandrine
%D 2005
%J Ann. Probab.
%K largest_eigenvalue random_matrices sample_covariance_matrix spectral_theory
%N 5
%P 1643--1697
%R 10.1214/009117905000000233
%T Phase transition of the largest eigenvalue for nonnull complex
sample covariance matrices
%U http://dx.doi.org/10.1214/009117905000000233
%V 33
%X 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.
@article{baik2005phase,
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.
},
added-at = {2013-12-13T20:55:09.000+0100},
author = {Baik, Jinho and Ben Arous, G{\'e}rard and P{\'e}ch{\'e}, Sandrine},
biburl = {https://www.bibsonomy.org/bibtex/2db6030d58ddab66fc5da7fca32f80c51/peter.ralph},
coden = {APBYAE},
doi = {10.1214/009117905000000233},
fjournal = {The Annals of Probability},
interhash = {6832ac99de3a25ace49424dbf0b13af7},
intrahash = {db6030d58ddab66fc5da7fca32f80c51},
issn = {0091-1798},
journal = {Ann. Probab.},
keywords = {largest_eigenvalue random_matrices sample_covariance_matrix spectral_theory},
mrclass = {15A52 (60F99 62E20 62H20)},
mrnumber = {2165575 (2006g:15046)},
mrreviewer = {Florent Benaych-Georges},
number = 5,
pages = {1643--1697},
timestamp = {2013-12-13T20:55:09.000+0100},
title = {Phase transition of the largest eigenvalue for nonnull complex
sample covariance matrices},
url = {http://dx.doi.org/10.1214/009117905000000233},
volume = 33,
year = 2005
}