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.

本文介绍了两个新的场景中的 $ p	imes p $ 样本协方差矩阵的特征值的 Marchenko-Pastur 定理。 我们的主要技术工具是关于块独立坐标的随机变量的二次形式和随机张量的新的集中不等式。