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

Abstract

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 $\frac{n}{p}=y+O(n^{-1})\in (0,1]$. For those values of $n,p$ we investigate the rate of convergence of the expected spectral distribution function of the matrix $\mathbf{W}=\frac{1}{p}\mathbf{X}{\mathbf{X}}^{\ast }$ to the Marchenko-Pastur law with parameter $y$. Assuming the conditions $\mathbf{E}{X}_{jk}=0$, $\mathbf{E}{X}_{jk}^2=1$ and ${\sup }_{n,p\ge 1}{\sup }_{1\le j\le n,1\le k\le p}\mathbf{E}|X_{jk}{|}^4=:{\mu }_4<\infty ,{\sup }_{n,p\ge 1}{\sup }_{1\le j\le n,1\le k\le p}|X_{jk}|\le D{n}^{\frac{1}{4}},$ 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 $O(n^{-1})$.

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