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

TL;DR

This paper studies the fluctuations of resolvent traces of a tensor-based sample covariance matrix, showing they converge to a Gaussian distribution as matrix dimensions grow large.

Contribution

It extends previous work by analyzing the asymptotic Gaussian fluctuations of resolvent traces in a tensor model of sample covariance matrices.

Findings

Resolvent trace fluctuations converge to a 2D Gaussian distribution.

The convergence holds as matrix dimensions tend to infinity with a fixed ratio.

The covariance matrix of the limiting Gaussian is explicitly characterized.

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).