## Convergence rates of spectral distributions of large sample covariance matrices.(English)Zbl 1059.60036

Summary: We improve known results on the convergence rates of spectral distributions of large-dimensional sample covariance matrices of size $$p {\times} n$$. Using the Stieltjes transform, we first prove that the expected spectral distribution converges to the limiting Marchenko-Pastur distribution with the dimension sample size ratio $$y=y_{n}=p/n$$ at a rate of $$O(n^{- 1/2})$$ if $$y$$ keeps away from 0 and 1, under the assumption that the entries have a finite eighth moment. Furthermore, the rates for both the convergence in probability and the almost sure convergence are shown to be $$O_{\text{p}}(n^{-2/5})$$ and $$o_{\text{a.s.}}(n^{-2/5+\eta})$$, respectively, when $$y$$ is away from 1. It is interesting that the rate in all senses is $$O(n^{-1/8})$$ when $$y$$ is close to 1.

### MSC:

 60F15 Strong limit theorems 62H99 Multivariate analysis
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