A note on the fluctuations of the resolvent traces of a tensor model of sample covariance matrices

In this note, we consider a sample covariance matrix of the form $$M_n=\sum _{\alpha =1}^m{\tau }_{\alpha }{\mathbf{y}}_{\alpha }^{(1)}\otimes {\mathbf{y}}_{\alpha }^{(2)}({\mathbf{y}}_{\alpha }^{(1)}\otimes {\mathbf{y}}_{\alpha }^{(2)}{)}^T,$$ where $({\mathbf{y}}_{\alpha }^{(1)},{\mathbf{y}}_{\alpha }^{(2)}{)}_{\alpha }$ are independent vectors uniformly distributed on the unit sphere $S^{n-1}$ and ${\tau }_{\alpha }\in {\mathbb{R}}_{+}$. We show that as $m,n\to \mathrm{\infty }$, $m\mathrm{/}{n}^2\to c>0$, the centralized traces of the resolvents, $\mathrm{T}\mathrm{r}(M_n-z{I}_n{)}^{-1}-\mathbf{E}\mathrm{T}\mathrm{r}(M_n-z{I}_n{)}^{-1}$, $\mathrm{\Im }z\ge {\eta }_0>0$, converge in distribution to a two-dimensional Gaussian random variable with zero mean and a certain covariance matrix. This work is a continuation of Dembczak-Kołodziejczyk and Lytova (2023), and Lytova (2018).