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Convergence rates of eigenvector empirical spectral distribution of large dimensional sample covariance matrix. (English) Zbl 1285.15018
The eigenvector empirical spectral distribution (VESD) is adopted to investigate the limiting behavior of eigenvectors and eigenvalues of covariance matrices. The authors show that the Kolmogorov distance between the expected VESD of the 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 $$10$$th-moment condition of the underlying distribution. It is also shown that, for any fixed $$\nu>0$$, the convergence rates of VESD are $$O(N^{-1/4})$$ in probability and $$O(N^{-1/4+\nu})$$ almost surely, requiring the finite $$8$$th-moment of the underlying distribution.

##### MSC:
 15B52 Random matrices (algebraic aspects) 15A18 Eigenvalues, singular values, and eigenvectors 60F15 Strong limit theorems 62E20 Asymptotic distribution theory in statistics 60F17 Functional limit theorems; invariance principles 62J10 Analysis of variance and covariance (ANOVA)
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