Marchenko-Pastur law for tensor powers of exchangeable unconditional vectors

Abstract

Given an isotropic, exchangeable, and unconditional random vector $\mathbf X$, we consider the sample covariance matrix constructed from i.i.d. copies of several tensor models of $\mathbf X$, such as the tensor power $\mathbf{X}^{\otimes d}$. Under appropriate moment conditions on $\mathbf X$, we show that almost surely, the empirical spectral distribution converges weakly to the Marchenko-Pastur law. This extends previous results which required the coordinates of $\mathbf X$ to be independent. As we demonstrate, our extension applies to many new random vectors $\mathbf X$ of interest.