\left\{ E>0 : \int_{-\infty}^{E^2} \rho_{{\mathsf{MP}}}(x){\mathrm{d}}x {\geqslant}\dfrac{k}{N} \right\},\ \ \ 1 {\leqslant}k {\leqslant}N.$$ Following the symmetrization trick, we also define $ \gamma_{-k} = -\gamma_k $. By a change of variable, it is easy to check that $$\label{e.typical}
\int_{-\infty}^{\gamma_k} \rho(x){\mathrm{d}}x=\dfrac{N+k}{2N}, \ \ \ \int_{-\infty}^{\gamma_{-k}} \rho(x){\mathrm{d}}x = \dfrac{N-k}{2N},$$ where $ \rho(x) $ is the counterpart of Marchenko-Pastur law for singular values, defined by $$\label{e.measure}
\rho(x)=\dfrac{1}{2\pi \xi}\sqrt{\dfrac{[(x^2-\lambda_-)(\lambda_+ - x^2)]_+}{x^2}},\ \ \ \sqrt{\lambda_-} {\leqslant}|x| {\leqslant}\sqrt{\lambda_+}.$$

Denote $ s_1 {\leqslant}\cdots {\leqslant}s_N $ the singular values of the data matrix $ X $, and extend the singular values following the symmetrization trick by $ s_{-k}=-s_k $. For $ z=E+{\text{\rm i}}\eta \in \mathbb{C} $ with $ \eta>0 $, let $ m_N(z) $ and $ S_N(z) $ denote the Stieltjes transform of the empirical measure of the (symmetrized) singular values and eigenvalues, respectively: $$m_N(z):= \dfrac{1}{2N}\sum_{-N {\leqslant}k {\leqslant}N} \dfrac{1}{s_k -z},\ \ \ S_N(z) := \dfrac{1}{N}\sum_{k=1}^N \dfrac{1}{\lambda_k -z}.$$ As mentioned previously, in the summation from $ -N $ to $ N $ the $ 0 $ index is always excluded. Note that due to the symmetrization, this is equivalent to $$\label{e.DiscreteStieltjes}
 m_N(z) = \dfrac{1}{2N}\sum_{k=1}^N \left( \dfrac{1}{s_k -z} + \dfrac{1}{-s_k - z} \right) = \dfrac{1}{N} \sum_{k=1}^N \dfrac{z}{s_k^2 - z^2} = z S_N(z^2).$$ 