On asymptotics of eigenvectors of large sample covariance matrix

By Z. Bai, B. Miao, Guangming Pan. 2007

154 citations recorded by the source corpus.

Let {X ij }, i, j = ..., be a double array of i.i.d. complex random variables with EX 11 = 0, E|X 11 | 2 = 1 and E|X 11 | 4 <∞, and let An = (1 N T 1/2 n X n X* n (T 1/2 n , where T 1/2 n is the square root of a nonnegative definite matrix T n and X n is the n x N matrix of the upper-left comer of the double array. The matrix An 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 An, 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 An are proved to have Gaussian limits, which suggests that the eigenvector matrix of An is nearly Haar distributed when T n is a multiple of the identity matrix, an easy consequence for a Wishart matrix.

Subfields: Computational Theory and Mathematics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/9f90314cce95675657e2b9da622fb5d7e6ea6e81

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Citations in the corpus

Sparse regular random graphs: Spectral density and eigenvectors (2009, background): “Traditionally, there has been much less work on computing the actual graph eigenvector distributions, with the notable and recent exception of Wishart-like sample covariance matrices (see Bai, Miao, and Pan [4]).”
Random matrix theory in statistics: A review (2014, background): “Bai et al. (2007) extended the results of Silverstein (1989, 1990) for matrices of the form n 1A1=2p XnX n nA 1=2 p where Ap is a p p nonnegative definite matrix and the p n data matrix Xn has i.i.d. standardized entries.”
Eigenvectors of some large sample covariance matrix ensembles (2009, unknown): “Our functionΦN can be compared with the object introduced in [3]: ∀λ ∈ R, FN 1 (λ ) = ∑i=1 |ui xN|(2) 1[λi ,+∞)(λ ), where(xN)N=1,2,.”
ON BILINEAR FORMS BASED ON THE RESOLVENT OF LARGE RANDOM MATRICES (2010, background): “16)] where such properties are established in the centered Wigner case to describe fine properties of the spectrum) , Central Limit Theorems [17, 13], behavior of the extreme eigenvalues of ΣnΣ ∗ n, behavior of the eigenvalues and eigenvectors associated with finite rank perturbations of ΣnΣ ∗ n [6]”
The spectrum of kernel random matrices (2010, background): “In particular, we refer the reader to [4, 5, 8, 17, 19, 21, 27, 30, 31, 37, 42] and [45] for some of the most statistically relevant results in this area.”
Spectral convergence for a general class of random matrices (2011, background): “In particular, spectral functions of the form of (3) or, equivalently, (4) were considered in [1] (see also references therein) to study the limiting eigenvectors of sample covariance matrices.”
Universal spectrum of normal modes in low-temperature glasses (2015, methodology): “Also, the eigenvectors of Wishart matrices are delocalized [36] and are asymptotically distributed according to the uniform Haar measure on the sphere.”
Central limit theorem for signal-to-interference ratio of reduced rank linear receiver (2008, methodology): “To prove the preceding theorems, we develop a central limit theorem for the eigenvalues and eigenvectors of the sample covariance matrices, which is a supplement of Theorem 2 in Bai, Miao and Pan [5].”