Open Access
August 2015 Convergence of the empirical spectral distribution function of Beta matrices
Zhidong Bai, Jiang Hu, Guangming Pan, Wang Zhou
Bernoulli 21(3): 1538-1574 (August 2015). DOI: 10.3150/14-BEJ613


Let $\mathbf{B}_{n}=\mathbf{S}_{n}(\mathbf{S}_{n}+\alpha_{n}\mathbf{T}_{N})^{-1}$, where $\mathbf{S}_{n}$ and $\mathbf{T}_{N}$ are two independent sample covariance matrices with dimension $p$ and sample sizes $n$ and $N$, respectively. This is the so-called Beta matrix. In this paper, we focus on the limiting spectral distribution function and the central limit theorem of linear spectral statistics of $\mathbf{B}_{n}$. Especially, we do not require $\mathbf{S}_{n}$ or $\mathbf{T}_{N}$ to be invertible. Namely, we can deal with the case where $p>\max\{n,N\}$ and $p<n+N$. Therefore, our results cover many important applications which cannot be simply deduced from the corresponding results for multivariate $F$ matrices.


Download Citation

Zhidong Bai. Jiang Hu. Guangming Pan. Wang Zhou. "Convergence of the empirical spectral distribution function of Beta matrices." Bernoulli 21 (3) 1538 - 1574, August 2015.


Received: 1 August 2012; Revised: 1 November 2013; Published: August 2015
First available in Project Euclid: 27 May 2015

zbMATH: 1319.60006
MathSciNet: MR3352053
Digital Object Identifier: 10.3150/14-BEJ613

Keywords: Beta matrices , CLT , LSD , multivariate statistical analysis

Rights: Copyright © 2015 Bernoulli Society for Mathematical Statistics and Probability

Vol.21 • No. 3 • August 2015
Back to Top