## Bernoulli

• Bernoulli
• Volume 21, Number 3 (2015), 1538-1574.

### Convergence of the empirical spectral distribution function of Beta matrices

#### Abstract

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.

#### Article information

Source
Bernoulli, Volume 21, Number 3 (2015), 1538-1574.

Dates
Revised: November 2013
First available in Project Euclid: 27 May 2015

https://projecteuclid.org/euclid.bj/1432732029

Digital Object Identifier
doi:10.3150/14-BEJ613

Mathematical Reviews number (MathSciNet)
MR3352053

Zentralblatt MATH identifier
1319.60006

#### Citation

Bai, Zhidong; Hu, Jiang; Pan, Guangming; Zhou, Wang. Convergence of the empirical spectral distribution function of Beta matrices. Bernoulli 21 (2015), no. 3, 1538--1574. doi:10.3150/14-BEJ613. https://projecteuclid.org/euclid.bj/1432732029

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