Convergence rates of eigenvector empirical spectral distribution of large dimensional sample covariance matrix
Ningning Xia, Yingli Qin, Zhidong Bai
The Annals of Statistics · 2013 · Volume 41 · Issue 5
Abstract
The eigenvector Empirical Spectral Distribution (VESD) is adopted to investigate the limiting behavior of eigenvectors and eigenvalues of covariance matrices. In this paper, we shall show that the Kolmogorov distance between the expected VESD of sample covariance matrix and the Marčenko–Pastur distribution function is of order $O(N^{-1/2})$. Given that data dimension $n$ to sample size $N$ ratio is bounded between 0 and 1, this convergence rate is established under finite 10th moment condition of the underlying distribution. It is also shown that, for any fixed $\eta>0$, the convergence rates of VESD are $O(N^{-1/4})$ in probability and $O(N^{-1/4+\eta})$ almost surely, requiring finite 8th moment of the underlying distribution.
Keywords
15A52, 60F15, 60F17, 62E20, 62H99, Eigenvector empirical spectral distribution, Empirical spectral distribution, Marčenko–Pastur distribution, Sample covariance matrix, Stieltjes transform
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