Compute the limit of trace of inverse of square of rank-1 perturbation of Wishart matrix

Question

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 $\rho$.

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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 $b,c>0$ ?

Answer 1

Let me try to answer the updated question. If $A$ and $B$ are Hermitian $m\times m$ matrices with $B$ of rank $r$ then the two sequences of eigenvalues $\lambda_k(A+B)$ of $A+B$ and $\lambda_k(A)$ of $A$, each sorted in ascending order, are related by $$\lambda_k(A)\leq\lambda_{k+r}(A+B)\leq\lambda_{k+2r}(A),\;\;1\leq k\leq m-2r.$$ For a proof see Theorem 8 of these notes. In this case $A=S(a)$ differs from $A+B=S(0)$ by a rank-one matrix, $r=1$, so the eigenvalues of $S(a)$ are interlaced with those of $S(0)$. Their spectral densities in the large-$m$ limit must coincide and Question 2 can be answered in the affirmative.