Isotropic local laws for sample covariance and generalized Wigner matrices
Isotropic local laws for sample covariance and generalized Wigner matrices
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DOI：
10.1214/ejp.v19-3054
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发表时间：
2014-03-15
影响因子：
1.4
通讯作者：
Yin, Jun
中科院分区：
文献类型：
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作者：
Bloemendal, Alex;Erdos, Laszlo;Yin, Jun
我们考虑X*X形式的样本协方差矩阵，其中X是具有独立随机项的M×N矩阵。我们证明了各向同性局部Marchenko-Pastur定律，即证明了预解(X*X-z)(-1)在二次型意义下收敛于恒等式的倍数。更确切地说，我们在量<；v，(X*X-z)(-1)w>；-v，w>；m(Z)上建立了精确的高概率界，其中m是Marchenko-Pastur定律的Stieltjes变换，v，w是C-N的一个元素。我们要求维度M和N的对数是可比较的。我们的结果保持在Imz>；=N-1+epsilon的标度范围内，并且在整个光谱范围内远离0。对于广义Wigner矩阵，我们也证明了类似的结果。
We consider sample covariance matrices of the form X*X, where X is an M x N matrix with independent random entries. We prove the isotropic local Marchenko-Pastur law, i.e. we prove that the resolvent (X*X - z)(-1) converges to a multiple of the identity in the sense of quadratic forms. More precisely, we establish sharp high-probability bounds on the quantity < v, (X*X - z)(-1)w > - < v, w > m(z), where m is the Stieltjes transform of the Marchenko-Pastur law and v, w is an element of C-N. We require the logarithms of the dimensions M and N to be comparable. Our result holds down to scales Im z >= N-1+epsilon and throughout the entire spectrum away from 0. We also prove analogous results for generalized Wigner matrices.