On spectrum of sample covariance matrices from large tensor vectors

In this paper, we investigate the limiting spectral distribution (LSD) of sums of independent rank-one matrices obtained from k-fold tensor products of n-dimensional vectors as k, n -+ oo. Assuming that the base vectors are complex random variables with unit modular, we show that the LSD is the Mar & ccaron;enko-Pastur law. Comparing with the existing results, our limiting setting allows k to grow much faster than n. Consequently, we obtain the necessary and sufficient conditions for Mar & ccaron;enko-Pastur law to serve as the LSD of our matrix model. Our approach is based on the moment method.