A note on the fluctuations of the resolvent traces of a tensor model of sample covariance matrices
Alicja Dembczak-Kołodziejczyk · arXiv (Cornell University) · 2024
In this note, we consider a sample covariance matrix of the form $$M_{n}=\sum_{α=1}^m τ_α{\mathbf{y}}_α^{(1)} \otimes {\mathbf{y}}_α^{(2)}({\mathbf{y}}_α^{(1)} \otimes {\mathbf{y}}_α^{(2)})^T,$$ where $(\mathbf{y}_α^{(1)},\, {\mathbf{y}}_α^{(2)})_α$ are independent vectors uniformly distributed on the unit sphere $S^{n-1}$ and $τ_α\in \mathbb{R}_+ $. We show that as $m, n \to \infty$, $m/n^2\to c>0$, the centralized traces of the resolvents, $\mathrm{Tr}(M_n-zI_n)^{-1}-\mathbf{E}\mathrm{Tr}(M_n-zI_n)^{-1}$, $\Im z\ge η_0>0$, 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).