大维样本协方差矩阵谱分布收敛速度的一个注记

提高了在有限八阶矩条件下,p×n维大维样本协方差矩阵谱分布收敛到Marcenko-Pastur分布的速度.特别,如果样本维数比率y=yn=p/n接近1,p×n维大维样本协方差矩阵谱分布的期望收敛到极限分布的速度,改进为O(n-1/6).相似在y接近1的条件下,依概率收敛和几乎处处收敛速度为Op(n-1/6)和Oa.s.(n-1/6).

A Note on the Convergence Rate of the Spectral Distributions of Large Sample Covariance Matrices

In this note, we improve the convergence rates of spectral distributions that large-dimensional sample covariance matrices of size p × n tend to the Marcenko-Pastur distribution, under the assumption that the entries have a finite eighth moment. Especially, we show that the expected spectral distributions of large-dimensional sample covariance matrices of size p × n tends to the limiting distribution with the dimension sample size ratio y = yn = p/n at the rate of O(n-1/6) if y is close to 1 . Similar results for both the convergence in probability and the almost sure convergence are shown to be Op(n-1/6) and Oa.s.(n-1/6), under the condition that y is close to 1.