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arXiv 1412.6284math.PR
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期望谱分布函数对Marchenko - Pastur定律收敛速率

Rate of Convergence of the Expected Spectral Distribution Function to the Marchenko -- Pastur Law

F. Götze, A. N. Tikhomirov

备注：
arXiv admin note: substantial text overlap with arXiv:1405.7820, arXiv:1407.2780
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摘要：

令$\mathbf X=(X_{jk})$表示具有对于$1\le j\le n, 1\le k\le p$独立的条目$X_{jk}$的$n\times p$随机矩阵。令$n,p$趋于无穷大，使得$\frac np=y+O(n^{-1})\in(0,1]$。对于这些值的$n,p$，我们调查的收敛速率的预期谱分布函数的矩阵$\mathbf W=\frac1{ p}\mathbf X\mathbf X^*$与参数$y$的马琴科-帕斯托尔定律。假设条件$\mathbf E X_{jk}=0$，$\mathbf E X_{jk}^2=1$和$ \quad \quad \quad \quad \quad \quad \quad \sup_{n,p\ge1}\sup_{1\le j\le n,1\le k\le p}\mathbf E |X_{jk}|^4=: μ_4<\infty,\quad \sup_{n,p\ge1} \sup_{1\le j\le n,1\le k\le p}|X_{jk}|\le D n^{\frac14},$，我们表明，期望的谱分布的样本协方差矩阵$\mathbf W$和Marchenko - Pastur定律之间的Kolmogorov距离是为了$O(n^{-1})$。

英文摘要：

Let $\mathbf X=(X_{jk})$ denote a $n\times p$ random matrix with entries $X_{jk}$, which are independent for $1\le j\le n, 1\le k\le p$. Let $n,p$ tend to infinity such that $\frac np=y+O(n^{-1})\in(0,1]$. For those values of $n,p$ we investigate the rate of convergence of the expected spectral distribution function of the matrix $\mathbf W=\frac1{ p}\mathbf X\mathbf X^*$ to the Marchenko-Pastur law with parameter $y$. Assuming the conditions $\mathbf E X_{jk}=0$, $\mathbf E X_{jk}^2=1$ and $ \quad \quad \quad \quad \quad \quad \quad \sup_{n,p\ge1}\sup_{1\le j\le n,1\le k\le p}\mathbf E |X_{jk}|^4=: μ_4<\infty,\quad \sup_{n,p\ge1} \sup_{1\le j\le n,1\le k\le p}|X_{jk}|\le D n^{\frac14},$ we show that the Kolmogorov distance between the expected spectral distribution of the sample covariance matrix $\mathbf W$ and the Marchenko -- Pastur law is of order $O(n^{-1})$.

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