# Central Limit Theorem for Linear Eigenvalue Statistics for a Tensor Product Version of Sample Covariance Matrices

Journal of Theoretical Probability, Jan 2017

For $k,m,n\in {\mathbb {N}}$, we consider $n^k\times n^k$ random matrices of the form \begin{aligned} {\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 \cdots \otimes {\mathbf {y}}_\alpha ^{(k)}, \end{aligned} 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 $m,n\rightarrow \infty$, $m/n^k\rightarrow c\in [0,\infty )$ 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 nonrandom limit found in Marchenko and Pastur (Math USSR Sb 1:457–483, 1967). For $k=2$, we define a subclass of good vectors ${\mathbf {y}}$ for which the centered linear eigenvalue statistics $n^{-1/2}{{\mathrm{Tr}}}\varphi ({\mathcal {M}}_{n,m,2}({\mathbf {y}}))^\circ$ converge in distribution to a Gaussian random variable, i.e., the Central Limit Theorem is valid.

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A. Lytova. Central Limit Theorem for Linear Eigenvalue Statistics for a Tensor Product Version of Sample Covariance Matrices, Journal of Theoretical Probability, 2017, 1-34, DOI: 10.1007/s10959-017-0741-9