The rate of convergence of spectra of sample covariance matrices

TL;DR

This paper establishes that the spectral distribution of a sample covariance matrix converges to the Marchenko-Pastur law at a rate of O(n^{-1/2}), uniformly across different matrix dimensions.

Contribution

It provides a uniform bound on the convergence rate of the spectral distribution of sample covariance matrices to the Marchenko-Pastur law.

Findings

Kolmogorov distance between spectral and Marchenko-Pastur distributions is O(n^{-1/2})

Bounds are uniform for all p, including p/n close to 1

Results apply to matrices with independent entries

Abstract

It is shown that the Kolmogorov distance between the spectral distribution function of a random covariance matrix $\frac1p XX^T$, 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 $\frac pn$ equal or close to 1.