论文标题
样品协方差合奏的硬边缘的本地Marchenko-Pastur法律
Local Marchenko-Pastur law at the hard edge of the Sample Covariance ensemble
论文作者
论文摘要
考虑一下$ n $ by $ n $矩阵$ x $的复杂条目，带有真实和虚构的零件。我们表明，$ x^*x $的本特征值的局部密度以最佳规模收敛于Marchenko-Pastur法律，概率为$ 1 $。我们还获得了散装和硬边和柔软边缘的特征值的刚性。在这里，我们通过直接与stieltjes变换的高预期能力直接工作来避免对数和多项式校正。我们假设条目具有有限的第四刻，并以$ n^{1/4} $截断。在这项工作中，我们简化并调整了Götze-Tikhomirov和Cacciapuoti-Maltsev-Schlein的先前论文中的方法。
Consider an $N$ by $N$ matrix $X$ of complex entries with iid real and imaginary parts. We show that the local density of eigenvalues of $X^*X$ converges to the Marchenko-Pastur law on the optimal scale with probability $1$. We also obtain rigidity of the eigenvalues in the bulk and near both hard and soft edges. Here we avoid logarithmic and polynomial corrections by working directly with high powers of expectation of the Stieltjes transforms. We work under the assumption that the entries have a finite 4th moment and are truncated at $N^{1/4}$. In this work we simplify and adapt the methods from prior papers of Götze-Tikhomirov and Cacciapuoti-Maltsev-Schlein to covariance matrices.
