Quantitative spectral rate for the normal-scores covariance

Let $\widehat{\bm{S}}_n$ be the normal-scores covariance, and let $m_n$ and $m_F$ denote its empirical Stieltjes transform and the limiting generalized Marčenko--Pastur Stieltjes transform, respectively. Assume --. Quantitative spectral rate. There is a constant $C>0$ such that, for every fixed $\eta_0>0$,

$$\sup_{\Im z\ge\eta_0}|m_n(z)-m_F(z)|=O_{\mathbb{P}}(n^{-1/2}),$$

and, more generally, a local Marčenko--Pastur law holds for $\widehat{\bm{S}}_n$ down to spectral scales $\Im z\gtrsim n^{-1+\varepsilon}$. Establishing this would require an anisotropic/local-law analysis tracking the rank-perturbation error inside the resolvent; this rate is not established by the results stated in the paper.