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

Abstract: 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 $m, n \to \infty$, $m/n^2\to c>0$, the centralized traces of the resolvents, $\mathrm{Tr}(M_n-zI_n)^{-1}-\mathbf{E}\mathrm{Tr}(M_n-zI_n)^{-1}$, $\Im z\ge \eta_0>0$, 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-Ko{\l}odziejczyk and Lytova (2023), and Lytova (2018).