On spectrum of sample covariance matrices from large tensor vectors

In this paper, we investigate the limiting empirical spectral distribution (LSD) of sums of independent rank-one $k$-fold tensor products of $n$-dimensional vectors as $k,n\to \mathrm{\infty }$. Assuming that the base vectors are complex random variables with unit modular, we show that the LSD is the Marčenko-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 Marčenko-Pastur law to serve as the LSD of our matrix model. Our approach is based on the moment method.