For sample covariance matrices associated with random vectors having graph -dependent entries and a number of dimensions growing with the sample size, we derive sharp con-ditions for the limiting spectrum of the matrices to have the same form as in the case of Gaussian data with similar covariance structure. Our results are tight. In particular, they give necessary and sufficient conditions for the Marchenko-Pastur theorem for sample covariance matrices associated with random vectors having m-dependent orthonormal elements when m = o(n).
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机译：对于与随机向量相关的样本协方差矩阵，这些向量具有与图相关的条目，并且随着样本量的增加而增长，我们推导出矩阵的极限谱具有与具有相似协方差结构的高斯数据相同的形式的尖锐条件。我们的结果很紧凑。特别是，当 m = o（n） 时，它们为样本协方差矩阵的 Marchenko-Pastur 定理提供了必要和充分的条件，该协方差矩阵与具有 m 依赖性正交元素的随机向量相关联。
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