On spectral distribution of sample covariance matrices from large dimensional and large k-fold tensor products

Abstract

We study the eigenvalue distributions for sums of independent rank-one k-fold tensor products of large n-dimensional vectors. Previous results in the literature assume that $k=o(n)$ and show that the eigenvalue distributions converge to the celebrated Marčenko-Pastur law under appropriate moment conditions on the base vectors. In this paper, motivated by quantum information theory, we study the regime where k grows faster, namely $k=O(n)$. We show that the moment sequences of the eigenvalue distributions have a limit, which is different from the Marčenko-Pastur law, and the Marčenko-Pastur law limit holds if and only if $k=o(n)$ for this tensor model. The approach is based on the method of moments.