LARGE COMPLEX CORRELATED WISHART MATRICES: FLUCTUATIONS AND ASYMPTOTIC INDEPENDENCE AT THE EDGES

• 成果类型: Article
• 署名作者: Hachem, Walid; Hardy, Adrien; Najim, Jamal
• 署名单位: IMT - Institut Mines-Telecom; Institut Polytechnique de Paris; Telecom Paris; Centre National de la Recherche Scientifique (CNRS); Royal Institute of Technology; Universite Gustave-Eiffel; ESIEE Paris; Institut Polytechnique de Paris; Ecole des Ponts ParisTech
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/15-AOP1022
• 发表日期: 2016
• 页码: 2264-2348
• 关键词: limiting spectral distribution; level-spacing distributions; sample covariance matrices; tracy-widom limit; LARGEST EIGENVALUE; PHASE-TRANSITION; UNIVERSALITY; statistics

摘要

We study the asymptotic behavior of eigenvalues of large complex correlated Wishart matrices at the edges of the limiting spectrum. In this setting, the support of the limiting eigenvalue distribution may have several connected components. Under mild conditions for the population matrices, we show that for every generic positive edge of that support, there exists an extremal eigenvalue which converges almost surely toward that edge and fluctuates according to the Tracy-Widom law at the scale N-2/3. Moreover, given several generic positive edges, we establish that the associated extremal eigenvalue fluctuations are asymptotically independent. Finally, when the leftmost edge is the origin ( hard edge), the fluctuations of the smallest eigenvalue are described by mean of the Bessel kernel at the scale N-2.