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Convergence of eigenvector empirical spectral distribution of sample covariance matrices. (arXiv:1705.03954v2 [math.PR] UPDATED)
来源于:arXiv
The eigenvector empirical spectral distribution (VESD) is a useful tool in
studying the limiting behavior of eigenvalues and eigenvectors of covariance
matrices. In this paper, we study the convergence rate of the VESD of sample
covariance matrices to the Marchenko-Pastur (MP) distribution. Consider sample
covariance matrices of the form $XX^*$, where $X=(x_{ij})$ is an $M\times N$
random matrix whose entries are independent (but not necessarily identically
distributed) random variables with mean zero and variance $N^{-1}$. We show
that the Kolmogorov distance between the expected VESD and the MP distribution
is bounded by $N^{-1+\epsilon}$ for any fixed $\epsilon>0$, provided that the
entries $\sqrt{N}x_{ij}$ have uniformly bounded 6th moment and that the
dimension ratio $N/M$ converges to some constant $d\ne 1$. This result improves
the previous one obtained in [33], which gives the convergence rate
$O(N^{-1/2})$ assuming $i.i.d.$ $X$ entries, bounded 10th moment and $d>1$.
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