Let $A_1=A_1(d)$, $A_2=A_2(d)$, and $B=B(d)$ be (sequences of) deterministic positive-definite $d \times d$ matrices and let $X_1$ and $X_2$ be independent random $n \times d$ matrices such that $X_k$ has iid rows from $N(0,A_k)$. Let $R$ be the resolvent of the (recaled) empirical covariance matrix $S := X_1^\top X_1 + X_2^\top X_2$, given by $R(z) := (S-z I_d)^{-1}$ for any $z \in \mathbb C^+$, and define $m(z) := \mathbb E[\mbox{tr}(BR(z))]$.

Let $\gamma \in (0,\infty)$ be fixed.

Question. Assume that $A_1$, $A_2$, and $B$ have limiting spectral distributions as $d \to \infty$. In the limit $n,d \to \infty$ such that $d/n \to \phi$, what is the value of $m(z)$ ?

Notes

• I'm only interested in computations for small $z$, i.e $z \to 0$.
• Observe that $S$ has the same distribution as $X^\top X$, where $X$ is a random $n \times d$ matrix with iid rows drawn from $N(0,A)$, for $A:= (A_1+A_2)/2$.
• If it helps, it may also be assumed that certain rational expressions of the matrices $A_1$, and $A_2$, and $B$, for example $BA^{-1}$, have limiting spectral densities.

Solution for the case $\phi \lt 1$

WLOG, assume $n \ge d + 2$. Then, it is a standard result that $$ \mathbb E R(0) = \mathbb E (X^\top X)^{-1} = \frac{1}{n-d-1}A^{-1}. $$

We deduce that $$ m(0) = \frac{1}{n-d-1}\mbox{tr}(B A^{-1}) = \frac{d}{n-d-1}\overline{\mbox{tr}}(B A^{-1}) \to \frac{\phi}{1-\phi}\cdot\lim_{d \to \infty} \overline{\mbox{tr}}(BA^{-1}), $$

where $\overline{\mbox{tr}} := (1/d)\mbox{tr}$ is the normalized trace operator.