Central limit theorem for linear spectral statistics of general separable sample covariance matrices with applications

Abstract

In this paper, we consider the separable covariance model, which plays an important role in wireless communications and spatio-temporal statistics and describes a process where the time correlation does not depend on the spatial location and the spatial correlation does not depend on time. We established a central limit theorem for linear spectral statistics of general separable sample covariance matrices in the form of ${\mathbf{S}}_n=\frac{1}{n}{\mathbf{T}}_{1n}{\mathbf{X}}_n{\mathbf{T}}_{2n}{\mathbf{X}}_n^{\ast }{\mathbf{T}}_{1n}^{\ast }$ where ${\mathbf{X}}_n=(x_{jk})$ is of $m_1\times m_2$ dimension, the entries $\{x_{jk},j=1,...,m_1,k=1,...,m_2\}$ are independent and identically distributed complex variables with zero means and unit variances, ${\mathbf{T}}_{1n}$ is a $p\times m_1$ complex matrix and ${\mathbf{T}}_{2n}$ is an $m_2\times m_2$ Hermitian matrix. We then apply this general central limit theorem to the problem of testing white noise in time series.

66 pages

References in corpus (5)

• Central limit theorem for linear eigenvalue statistics of random matrices with independent entries
• Central Limit Theorem for linear eigenvalue statistics of the Wigner and sample covariance random matrices
• Distribution functions for largest eigenvalues and their applications
• CLT for linear spectral statistics of large dimensional sample covariance matrices with dependent data
• No eigenvalues outside the limiting support of the spectral distribution of general sample covariance matrices