• 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

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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

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

Received: July 2017
Revised: January 2018
First available in Project Euclid: 12 June 2019

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


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.

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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.