Wangjun Yuan: 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 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 Marcenko-Pastur law, and 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.

This is a joint work with Benoit Collins and Jianfeng Yao.