On asymptotics of eigenvectors of large sample covariance matrix
On asymptotics of eigenvectors of large sample covariance matrix
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DOI：
10.1214/009117906000001079
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发表时间：
2007-07-01
影响因子：
2.3
通讯作者：
Pan, G. M.
中科院分区：
文献类型：
--
作者：
Bai, Z. D.;Miao, B. Q.;Pan, G. M.
设{X-ij}， i, j = ....是一个由iid个复随机变量组成的双数组，其中EX11 =0， E竖条x -11竖条(2)= 1，E竖条x -11竖条(4)<∞，设a (n) = 1/ n T-n(1/2) x -n(XnTn1/2)- t -*，其中T-n（1/2）为非负定矩阵T-n的平方根，x -n为双数组左上角的n × n矩阵。矩阵A(n)可以认为是来自均值为零且协方差矩阵为T-n的总体的iid样本的样本协方差矩阵，如果T-n是另一个样本协方差矩阵的逆，则可以认为是多元F矩阵。为了研究A(n)的特征向量的极限行为，定义了一种由特征向量定义权值的新形式的经验谱分布，并证明了它与由等权值定义的经验谱分布具有相同的极限谱分布。此外，如果{X-ij}和T-n是实数或复数，并做出一些额外的矩假设，则证明了由A(n)的特征向量定义的线性谱统计量具有高斯极限，这表明当T-n是单位矩阵的倍数时，A(n)的特征向量矩阵接近Haar分布，这是Wishart矩阵的一个简单结果。
Let {X-ij}, i, j = .... be a double array of i.i.d. complex random variables with EX11 =0, E vertical bar X-11 vertical bar(2) = 1 and E vertical bar X-11 vertical bar(4) < infinity, and let A(n) = 1/N T-n(1/2) X-n (XnTn1/2)-T-*, where T-n(1/2) is the square root of a nonnegative definite matrix T-n and X-n is the n x N matrix of the upper-left corner of the double array. The matrix A(n) can be considered as a sample covariance matrix of an i.i.d. sample from a population with mean zero and covariance matrix T-n, or as a multivariate F matrix if T-n is the inverse of another sample covariance matrix. To investigate the limiting behavior of the eigenvectors of A(n), a new form of empirical spectral distribution is defined with weights defined by eigenvectors and it is then shown that this has the same limiting spectral distribution as the empirical spectral distribution defined by equal weights. Moreover, if {X-ij} and T-n are either real or complex and some additional moment assumptions are made then linear spectral statistics defined by the eigenvectors of A(n) are proved to have Gaussian limits, which suggests that the eigenvector matrix of A(n) is nearly Haar distributed when T-n is a multiple of the identity matrix, an easy consequence for a Wishart matrix.