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 $$ \mathcal{M}{n,m,k}(\mathbf{y})=\sum{\alpha=1}^m\tau_\alpha {Y_\alpha}Y_\alpha^T,\quad Y_\alpha=\mathbf{y}\alpha^{(1)}\otimes...\otimes\mathbf{y}\alpha^{(k)}, $$ 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 $m,n\rightarrow \infty$, $m/n^k\rightarrow c\in \lbrack 0,\infty )$ 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 $n^{-1/2}\text{Tr} \,\varphi(\mathcal{M}{n,m,2}(\mathbf{y}))^\circ$ converge in distribution to a Gaussian random variable, i.e., the Central Limit Theorem is valid.