distribution. Early non-quantitative results were first proved in [@peche2009universality; @pillai2014universality; @Soshnikov]. Quantitative estimates, however, were only obtained for the Wishart ensemble (see [@el2006rate; @Johansson; @Johnstone; @ma2012accuracy]), which is essentially an integrable model. In this paper, we prove the explicit rate of convergence $ N^{-2/9} $ to the Tracy-Widom distribution for all sample covariance matrices of type $ X^*X $ with general distributed entries, and an analogous result with deterministic rate $ N^{-1/57} $ for all separable sample covariance matrices $ X^* \Sigma X $ with diagonal population $ \Sigma $. For simplicity and motivations from statistics, we only consider the real case, but the whole proof and the results are also true for complex sample covariance matrices.

Let $ X=(x_{ij}) $ be an $ M \times N $ data matrix with independent real valued entries with mean 0 and variance $ M^{-1} $, $$\label{e.Assumption1}
x_{ij} = M^{-1/2}q_{ij},\ \ \ {\mathbb{E}}q_{ij}=0,\ \ \ {\mathbb{E}}q_{ij}^2 =1.$$ Furthermore, we assume the entries $ q_{ij} $ have a sub-exponential decay, that is, there exists a constant $ \theta>0 $ such that for $ u>1 $, $$\label{e.Assumption2}
{\mathbb{P}}(|q_{ij}| > u) {\leqslant}\theta^{-1} \exp (-u^\theta).$$ This sub-exponential decay assumption is mainly for convenience, other conditions such as 