### 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 language | English (US) |
---|---|

Pages (from-to) | 1643-1697 |

Number of pages | 55 |

Journal | Annals of Probability |

Volume | 33 |

Issue number | 5 |

DOIs | |

State | Published - Sep 2005 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)
- Statistics and Probability

### Cite this

*Annals of Probability*,

*33*(5), 1643-1697. https://doi.org/10.1214/009117905000000233

**Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices.** / Baik, Jinho; Arous, Gérard Ben; Péché, Sandrine.

Research output: Contribution to journal › Article

*Annals of Probability*, vol. 33, no. 5, pp. 1643-1697. https://doi.org/10.1214/009117905000000233

}

TY - JOUR

T1 - Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices

AU - Baik, Jinho

AU - Arous, Gérard Ben

AU - Péché, Sandrine

PY - 2005/9

Y1 - 2005/9

N2 - 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.

AB - 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.

KW - Airy kernel

KW - Limit theorem

KW - Random matrix

KW - Sample covariance

KW - Tracy-Widom distribution

UR - http://www.scopus.com/inward/record.url?scp=27644476898&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=27644476898&partnerID=8YFLogxK

U2 - 10.1214/009117905000000233

DO - 10.1214/009117905000000233

M3 - Article

AN - SCOPUS:27644476898

VL - 33

SP - 1643

EP - 1697

JO - Annals of Probability

JF - Annals of Probability

SN - 0091-1798

IS - 5

ER -