The rate of convergence of spectra of sample covariance matrices

Abstract

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