CLT for linear eigenvalue statistics for a tensor product version of sample covariance matrices

Abstract

For $k,m,n\in \mathbb{N}$, we consider $n^k\times n^k$ random matrices of the form $${\mathcal{M}}_{n,m,k}(\mathbf{y})=\sum _{\alpha =1}^m{\tau }_{\alpha }{Y}_{\alpha }{Y}_{\alpha }^T,Y_{\alpha }={\mathbf{y}}_{\alpha }^{(1)}\otimes ...\otimes {\mathbf{y}}_{\alpha }^{(k)},$$ where ${\tau }_{\alpha }$, $\alpha \in [m]$, are real numbers and ${\mathbf{y}}_{\alpha }^{(j)}$, $\alpha \in [m]$, $j\in [k]$, are i.i.d. copies of a normalized isotropic random vector $\mathbf{y}\in {\mathbb{R}}^n$. For every fixed $k\ge 1$, if the Normalized Counting Measures of $\{{\tau }_{\alpha }{\}}_{\alpha }$ converge weakly as $m,n\to \infty$, $m/n^k\to c\in [0,\infty )$ and $\mathbf{y}$ is a good vector in the sense of Definition 1.1, then the Normalized Counting Measures of eigenvalues of ${\mathcal{M}}_{n,m,k}(\mathbf{y})$ converge weakly in probability to a non-random limit found in [15]. For $k=2$, we define a subclass of good vectors $\mathbf{y}$ for which the centered linear eigenvalue statistics $n^{-1/2}\text{Tr}\phi ({\mathcal{M}}_{n,m,2}(\mathbf{y}){)}^{\circ }$ converge in distribution to a Gaussian random variable, i.e., the Central Limit Theorem is valid.

References in corpus (7)

• Central limit theorem for linear eigenvalue statistics of random matrices with independent entries
• Entropy and Entanglement in Quantum Ground States
• Gaussian fluctuations for linear spectral statistics of large random covariance matrices
• Functional CLT for sample covariance matrices
• The universality principle for spectral distributions of sample covariance matrices
• The Central Limit Theorem for Linear Eigenvalue Statistics of the Sum of Independent Matrices of Rank One
• Distribution of eigenvalues of sample covariance matrices with tensor product samples