  We study the limiting spectral distribution of large-dimensional sample
covariance matrices associated with symmetric random tensors formed by
$\binom{n}{d}$ different products of $d$ variables chosen from $n$ independent
standardized random variables. We find optimal sufficient conditions for this
distribution to be the Marchenko-Pastur law in the case $d=d(n)$ and
$n\to\infty$. Our conditions reduce to $d^2=o(n)$ when the variables have
uniformly bounded fourth moments. The proofs are based on a new concentration
inequality for quadratic forms in symmetric random tensors and a law of large
numbers for elementary symmetric random polynomials.
