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

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 Mar\v{c}enko-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 Mar\v{c}enko-Pastur law. As a byproduct, we show that the Mar\v{c}enko-Pastur law limit holds if and only if $k=o(n)$ for this tensor model. The approach is based on the method of moments.