The Rate of Convergence of Spectra of Sample Covariance Matrices

It is shown that the Kolmogorov distance between the spectral distribution function of a random covariance matrix $p^{-1}XX^T$, where $X$ is an $n\times p$ matrix with independent entries and the distribution function of the Marchenko–Pastur law is of order $O(n^{-1/2})$. The bounds hold uniformly for any $p$, including $p/n$ equal or close to $1$.