Let $a \ge 0$, $b,c>0$ be fixed constants, and let $X$ be an $m \times d$ random matrix with entries drawn iid from $N(0,1/d)$. Consider the random psd matrix $S := a 1_m 1_m^\top + b XX^\top + c I_m$.

Question 1.In the limit $m,d \to \infty$ with $m/d \to \rho \in (0,\infty)$, what does $d^{-1}\mbox{trace}(S^{-2})$ converge to ?

**Observation.** If $a = 0$, then one computes
$$
d^{-1}\mbox{trace}(S^{-2}) = \frac{1}{d}\sum_{i=1}^n \frac{1}{(bXX^\top +c)^2} \overset{a.s}{\to} b^{-2}m'_{MP(\rho)}(-c/b),
$$
where $m_{MP(\rho)}$ is the Stieltjes transform of the Marchenko-Pastur distribution with parameter $\rho$.

## Update

I've often heard that

"

Finite-rank perturbations don't change limiting empirical spectral distribution of random matrices."

Unfortunately, I can't find a definitive reference for this statement.

Question 2.In view of the previous remark, is it true that $d^{-1}\mbox{trace}(S^{-2}) \overset{a.s}{\to} b^{-2}m'_{MP(\rho)}(-c/b)$ for all $a \ge 0$ and $b,c>0$ ?