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Work with $z$ in the upper half-plane. Write the Stieltjes matrix resolvents and their normalized matrix traces asThe minor trace is normalized by $N$, not by $N-1$. This sign convention is the negative of the convention $(zI-X_N{)}^{-1}$ used in the general resolvent of an operator article. Here $g$ is the Stieltjes transform of a measure of the empirical spectral measure, with kernel $(x-z{)}^{-1}$.

The diagonal entries of $X_N$ are zero, so the preceding Schur complement formula gives $G_{ii}=-1/(z+q_i)$. Taking the matrix trace, subtracting the comparison value $-1/(z+g)$, and combining fractions gives the resolvent self-consistency defectThe positive numerator sign is fixed by this subtraction. All denominators are nonzero in the upper half-plane, as the imaginary-part estimate in the next part shows. The identity is deterministic and does not use entry independence or moment assumptions.

By an orthonormal eigenbasis, the imaginary part of the Stieltjes matrix resolvent has eigenvalues $\eta /|\lambda -z{|}^2>0$. Thus its normalized trace has positive imaginary part, and its quadratic form at any vector has nonnegative imaginary part. This yields $|z+g|\ge \eta$ and $|z+q|\ge \eta$ for the normalized trace $g$ and a quadratic form $q$ used in Schur-complement estimates.

For a finite positive measure on the real line, this convention for its Stieltjes transform is analytic off the real line and has positive imaginary part in the upper half-plane if the measure is nonzero. Some sources use $(z-x{)}^{-1}$ instead, changing the sign. The transform of an empirical spectral measure equals the normalized trace of the Stieltjes matrix resolvent.