I found in the book "An introduction to random matrices", an exercise with which I am struggling and have not found an answer for:
Exercise 2.4.8:(p.50) Let $Y_N$ be a sequence of $N\times M(N)$ matrices with i.i.d entries with zero mean and variance $\frac{1}{N}$ such that $M(N)/N\rightarrow \alpha\in[1,\infty)$, and: $$ N^{k/2} \mathbb{E}\Big[ \big\vert [Y_N]_{1,1} \big\vert^k \Big] \leq r_k<\infty \quad \text{for all} \quad k,N \in \mathbb{N} $$ By writing $$ W_N=Y_N Y_N^T= \sum\limits_{i=1}^{M(N)} \mathbf{y}_i \mathbf{y}_i \quad \text{for appropriate vectors} $$ Show that $\frac{1}{N}\text{tr} \big[ \mathcal{S}_{W_N}(z) \big]$ , where $\mathcal{S}_{W_N}(z)=\big( W_N-zI_N \big)^{-1}$, converges to the solution of the equation: $$ m(z)= \dfrac{-1}{z- \frac{\alpha}{1+m(z)} }$$ Hint: Use the equality: $$ I_N+(z-x)\big[ W_N-zI_N \big]^{-1}=(W_N-xI_N)\big[ W_N-zI_N \big]^{-1} $$ and then use the equality: $$ \mathbf{y}_i^T\big( B_i+\mathbf{y}_i\mathbf{y}_i^T \big)^{-1}= \frac{1}{1+\mathbf{y}_i^T B_i^{-1} \mathbf{y}_i } \mathbf{y}_i^TB_i^{-1} \quad \text{for} \quad B_i=W_N-zI-\mathbf{y}_i\mathbf{y}_i^T $$ to show that the normalized trace of $ \text{tr}\Big( (W_N-xI_N)\big[ W_N-zI_N \big]^{-1} \Big) $ converges to $0$.
My attempt thus far: I think I understand how to solve the problem once I show that the above trace does tend to $0$. If I denote:
$$ \alpha_N(x,z):= \frac{1}{N}\text{tr}\Big( (W_N-xI_N)\big[ W_N-zI_N \big]^{-1} \Big) $$ Then the first equality means that $\alpha_N(x,z)=1+(z-x)\text{tr}\big(S_{W_N}(z)\big)$. The second equality yields that: $$ \sum\limits_{i=1}^{M(N)} \frac{\mathbf{y}_i^TB_i^{-1}\mathbf{y}_i}{1+\mathbf{y}_i^TB_i^{-1}\mathbf{y}_i} =\text{tr} \big(W_N S_{W_N}(z) \big)=N+z\text{tr}\big(S_{W_N}(z)\big) $$ Thanks to this I can write: $$\alpha_N(x,z)= \frac{1}{N} \sum\limits_{i=1}^{M(N)} \frac{\mathbf{y}_i^TB_i^{-1}\mathbf{y}_i}{1+\mathbf{y}_i^TB_i^{-1}\mathbf{y}_i}-\frac{x}{N}\text{tr}\big(S_{W_N}(z) \big) $$ But my attempts to play with this equality to show it does tend to $0$, have all seemed unsuccessful and messy.
So I would like to ask whether someone here encountered this exercise and solved it, or is familiar with the subject matter and suggest a hint how to proceed?