On asymptotics of eigenvectors of large sample covariance matrix

Details On asymptotics of eigenvectors of large sample covariance matrix On asymptotics of eigenvectors of large sample covariance matrix

2007-08-13

arxiv.org/abs/0708.1720v1

Annals of Probability 2007, Vol. 35, No. 4, 1532-1572

Mathematics

Probability

Published at http://dx.doi.org/10.1214/009117906000001079 in the Annals of Probability (http://www.imstat.org/aop/) by the Ins

Scientific paper

10.1214/009117906000001079

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 $E|X_{11}|^4<\infty$, and let $A_n=\frac{1}{N}T_n^{{1}/{2}}X_nX_n^*T_n^{{1}/{2}}$, where $T_n^{{1}/{2}}$ is the square root of a nonnegative definite matrix $T_n$ and $X_n$ is the $n\times N$ 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.

Affiliated with

Also associated with

No associations

Rating

On asymptotics of eigenvectors of large sample covariance matrix does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with On asymptotics of eigenvectors of large sample covariance matrix, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On asymptotics of eigenvectors of large sample covariance matrix will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-23640