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

• 成果类型: Article
• 署名作者: Baik, J; Ben Arous, G; Péché, S
• 署名单位: University of Michigan System; University of Michigan; Swiss Federal Institutes of Technology Domain; Ecole Polytechnique Federale de Lausanne
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117905000000233
• 发表日期: 2005
• 页码: 1643-1697
• 关键词: DISTRIBUTIONS; subsequences; asymptotics; edge

摘要

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.