We prove the Marchenko–Pastur law for the eigenvalues of p × 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 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(n1/3). Our main technical tools are new concentration inequalities for quadratic forms in random variables with block-independent coordinates, and for random tensors.

中文翻译：

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具有宽松独立条件的马尔琴科-巴斯德法

我们证明了特征值的 Marchenko-Pastur 定律p × p在数据没有独立坐标的两种新情况下采样协方差矩阵。在第一个场景中——块独立模型——p数据的坐标被划分为块，不同块中的条目是独立的，但来自同一块的条目可能是相互依赖的。在第二种情况下——随机张量模型——数据是有序的齐次随机张量d，即数据的坐标都是n d的不同产品d从一组变量中选择n独立随机变量。我们表明，只要最大块的大小为○(p)，对于随机张量模型，只要d = ○(n1/3). 我们的主要技术工具是用于具有块独立坐标的随机变量中的二次形式和随机张量的新浓度不等式。