## Bernoulli

- Bernoulli
- Volume 25, Number 3 (2019), 1838-1869.

### Central limit theorem for linear spectral statistics of large dimensional separable sample covariance matrices

Zhidong Bai, Huiqin Li, and Guangming Pan

#### Abstract

Suppose that $\mathbf{X}_{n}=(x_{jk})$ is $N\times n$ whose elements are independent complex variables with mean zero, variance 1. The separable sample covariance matrix is defined as $\mathbf{B}_{n}=\frac{1}{N}\mathbf{T}_{2n}^{1/2}\mathbf{X}_{n}\mathbf{T}_{1n}\mathbf{X}_{n}^{*}\mathbf{T}_{2n}^{1/2}$ where $\mathbf{T}_{1n}$ is a Hermitian matrix and $\mathbf{T}_{2n}^{1/2}$ is a Hermitian square root of the nonnegative definite Hermitian matrix $\mathbf{T}_{2n}$. Its linear spectral statistics (LSS) are shown to have Gaussian limits when $n/N$ approaches a positive constant under some conditions.

#### Article information

**Source**

Bernoulli, Volume 25, Number 3 (2019), 1838-1869.

**Dates**

Received: July 2017

Revised: January 2018

First available in Project Euclid: 12 June 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1560326430

**Digital Object Identifier**

doi:10.3150/18-BEJ1038

**Mathematical Reviews number (MathSciNet)**

MR3961233

**Zentralblatt MATH identifier**

07066242

**Keywords**

central limit theorem linear spectral statistics random matrix theory separable sample covariance matrix

#### Citation

Bai, Zhidong; Li, Huiqin; Pan, Guangming. Central limit theorem for linear spectral statistics of large dimensional separable sample covariance matrices. Bernoulli 25 (2019), no. 3, 1838--1869. doi:10.3150/18-BEJ1038. https://projecteuclid.org/euclid.bj/1560326430

#### Supplemental materials

- Supplement to “Central limit theorem for linear spectral statistics of large dimensional separable sample covariance matrices”. In this supplement, we give four parts. Part A is the convergence of $M_{n2}(z)$. Part B shows the analysis of the remainder term for general case in Section 4. Some useful lemmas are listed in Part C and Part D.Digital Object Identifier: doi:10.3150/18-BEJ1038SUPPSupplemental files are immediately available to subscribers. Non-subscribers gain access to supplemental files with the purchase of the article.