Let $n$ and $d$ be positive integers with
$$
n,d \to \infty, \quad n/d \to \rho \in (0,\infty).
$$
Let $\Sigma_d$ be a psd matrix such that
- $\mbox{trace}(\Sigma_d) = 1$.
- $\|\Sigma_d\|_{op} = \mathcal O(1/d)$.
- The empirical eigenvalue distribution of $d \cdot \Sigma_d$ converges weakly to some distribution $D$ on $\mathbb R$. 

Let $W$ be a random $n \times d$ matrix with iid rows from $N(0,\Sigma_d)$. Finally, let $a,b \ge 0$ be fixed constants, and define random $A$ and $B$ by
$$
\begin{split}
A &= WW^\top + a I_n,\\
B &= WW^\top + b I_n.
\end{split}
$$

>**Question.** *What is an analytic formula for the limiting value of the (random) scalar $\dfrac{1_n^\top B^{-1}AB^{-1} 1_n}{d}$, where $1_n := (1,1,\ldots,1) \in \mathbb R^n$ ?*

Observations
---
If we replace $1_n$ by $z$, where $z \sim N(0,I_n)$ is independent of $W$, then
$$
\begin{split}
\mathbb E_z[z^\top B^{-1} A B^{-1} z/d] &= \mbox{trace}(B^{-1} A B^{-1}/d)\\
&= \frac{1}{d}\sum_{i=1}^n \frac{\lambda_i + a}{(\lambda_i + b)^2}\\
& \overset{a.s}{\to} \int_0^\infty \frac{\lambda + a}{(\lambda + b)^2}\,\mu(\mathrm{d}\lambda)\\
&= m_\mu(-b) - (a-b) m_\mu'(-b),
\end{split}
$$
where $\mu$ is the LSD of $WW^\top$, and $m_\mu$ is its Stieltjes transform (which can be evaluated via $D$ and Silverstein's fixed-point equation).

>*Maybe this computation is somehow linked to the an answer to my main question ?*