On spectrum of sample covariance matrices from large tensor vectors

In this paper, we study the limiting spectral distribution of sums of
independent rank-one large $k$-fold tensor products of large $n$-dimensional
vectors. In the literature, the limiting moment sequence is obtained for the
case $k=o(n)$ and $k=O(n)$. Under appropriate moment conditions on base
vectors, it has been showed that the eigenvalue empirical distribution
converges to the celebrated Mar\v{c}enko-Pastur law if $k=O(n)$ and the
components of base vectors have unit modulus, or $k=o(n)$. In this paper, we
study the limiting spectral distribution by allowing $k$ to grow much faster,
whenever the components of base vectors are complex random variables on the
unit circle. It turns out that the limiting spectral distribution is
Mar\v{c}enko-Pastur law. Comparing with the existing results, our limiting
setting only requires $k \to \infty$. Our approach is based on the moment
method.