of this paper, we consider the Kolmogorov distance $${\mathsf{d_K}}(X,Y) := \sup_{x} \left| {\mathbb{P}}(X {\leqslant}x) - {\mathbb{P}}(Y {\leqslant}x) \right|.$$

\[t.Rate\] Let $ H_N $ be sample covariance matrices satisfying and . Let $ {\mathsf{TW}}$ be the Tracy-Widom distribution. For any $ {\varepsilon}>0 $, for large enough $ N $ we have $${\mathsf{d_K}}(N^{2/3}(\lambda_N - \lambda_+),{\mathsf{TW}}) {\leqslant}N^{-\frac{2}{9}+{\varepsilon}}.$$

The null case $ X^*X $ is our primary concern in this paper, but quantitative estimates are also valid for general diagonal population matrices $ X^* \Sigma X $ thanks to the comparison theorem for the Green function flow by Lee and Schnelli (see Section \[s.GeneralPopulation\] for more details). Combining our quantitative edge universality for the null case (Theorem \[t.Rate\]) with the Green function comparison by Lee-Schnelli (Proposition \[p.GreenFunctionFlow\]), we derive the rate of convergence to Tracy-Widom distribution for separable sample covariance matrices with general diagonal population.

\[t.RateGeneralPopulation\] Let $ Q := X^* \Sigma X $ be an $ N \times N $ separable sample covariance matrix, where $ X $ is an $ M \times N $ real random matrix satisfying and , and $ \Sigma $ is a real diagonal $ M \times M $ matrix satisfying . Let $ \mu_N $ be the 