Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices

Jinho Baik, Gérard Ben Arous, Sandrine Péché

Research output: Contribution to journalArticle

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.

Original languageEnglish (US)
Pages (from-to)1643-1697
Number of pages55
JournalAnnals of Probability
Volume33
Issue number5
DOIs
StatePublished - Sep 2005

Keywords

  • Airy kernel
  • Limit theorem
  • Random matrix
  • Sample covariance
  • Tracy-Widom distribution

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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