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 $\frac np=y+O(n^{-1})\in(0,1]$. 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=: \mu_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 $O(n^{-1})$.