On spectral distribution of sample covariance matrices from large dimensional and large k-fold tensor products
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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. As a byproduct, we show that 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.
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