  We prove the Marchenko-Pastur law for the eigenvalues of $p \times p$ sample
covariance matrices in two new situations where the data does not have
independent coordinates. In the first scenario - the block-independent model -
the $p$ coordinates of the data are partitioned into blocks in such a way that
the entries in different blocks are independent, but the entries from the same
block may be dependent. In the second scenario - the random tensor model - the
data is the homogeneous random tensor of order $d$, i.e. the coordinates of the
data are all $\binom{n}{d}$ different products of $d$ variables chosen from a
set of $n$ independent random variables. We show that Marchenko-Pastur law
holds for the block-independent model as long as the size of the largest block
is $o(p)$ and for the random tensor model as long as $d = o(n^{1/3})$. Our main
technical tools are new concentration inequalities for quadratic forms in
random variables with block-independent coordinates, and for random tensors.
