SINGULAR VECTOR DISTRIBUTION OF SAMPLE COVARIANCE MATRICES
SINGULAR VECTOR DISTRIBUTION OF SAMPLE COVARIANCE MATRICES
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DOI：
10.1017/apr.2019.10
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发表时间：
2019-03-01
影响因子：
1.2
通讯作者：
Ding, Xiucai
中科院分区：
文献类型：
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作者：
Ding, Xiucai
考虑一类样本协方差矩阵Q = TXX*T*，其中X=（x（ij））是由独立同分布元素组成的M × N矩形矩阵，T是确定性矩阵，使得T*T是对角矩阵.假设M与N可比较，证明了当x（ij）的前两阶矩与高斯随机变量相一致时，右奇异向量中靠近边缘奇异值的分量的分布与高斯系综的分布一致.对于与体奇异值相关联的右奇异向量，如果x（ij）的前四阶矩与高斯随机变量的前四阶矩相匹配，则同样的结论成立。如果我们进一步假设T是对角的，则类似的结果对于左奇异向量也成立。
We consider a class of sample covariance matrices of the form Q = TXX*T*, where X= (x(ij)) is an M x N rectangular matrix consisting of independent and identically distributed entries, and T is a deterministic matrix such that T*T is diagonal. Assuming that M is comparable to N, we prove that the distribution of the components of the right singular vectors close to the edge singular values agrees with that of Gaussian ensembles provided the first two moments of x(ij) coincide with the Gaussian random variables. For the right singular vectors associated with the bulk singular values, the same conclusion holds if the first four moments of x(ij) match those of the Gaussian random variables. Similar results hold for the left singular vectors if we further assume that T is diagonal.