You can rewrite $$ S=\frac{1}{M}C^{1/2}XX^T(C^{1/2})^T $$ where $X_{.j}\sim \mathcal{N}(\boldsymbol 0, \boldsymbol I)$. It is then classical that the limiting eigenvalue distribution of $S$ (in the almost sure weak convergence) is given by $MP_\gamma\boxtimes\nu$, the free multiplicative convolution of the Marchenko-Pastur distribution $MP_\gamma$ of paramter $\gamma$ with the limiting eigenvalue distribution $\nu$ of $C$. Thus, you indeed obtain $MP_\gamma$ if and only if $\nu=\delta_1$, namely if and only if $$ \frac{1}{N}\sum_{i=1}^N\delta_{v_i} \longrightarrow \delta_0\qquad \mbox{weakly as $N\rightarrow\infty$ }. $$ For a reference, you can look at the book of Anderson, Guionnet and Zeitouni, "Introduction to random matrices" (Chapter 5), but also any introduction to free probability I guess.

As you can see, free probability provides a powerful language to describe such perturbed random matrix models!

You can rewrite $$ S=\frac{1}{M}C^{1/2}XX^T(C^{1/2})^T $$ where $X_{.j}\sim \mathcal{N}(\boldsymbol 0, \boldsymbol I)$. It is then classical that the limiting eigenvalue distribution of $S$ (in the almost sure weak convergence) is given by $MP_\gamma\boxtimes\nu$, the free multiplicative convolution of the Marchenko-Pastur distribution $MP_\gamma$ of paramter $\gamma$ with the limiting eigenvalue distribution $\nu$ of $C$. Thus, you indeed obtain $MP_\gamma$ if and only if $\nu=\delta_1$, namely if and only if $$ \frac{1}{N}\sum_{i=1}^N\delta_{v_i} \longrightarrow \delta_0\qquad \mbox{weakly as $N\rightarrow\infty$ }. $$ For a reference, you can look at the book of Anderson, Guionnet and Zeitouni, "Introduction to random matrices" (Chapter 5), but also any introduction to free probability I guess.

As you can see, free probability provides a powerful language to describe such perturbed random matrix models!

NB : As an example, a spiked model is the situation where only a fixed finite number of $v_i$'s are non zero, which fit with the above characterization.

You can rewrite $$ S=\frac{1}{M}C^{1/2}XX^T(C^{1/2})^T $$ where $X_{.j}\sim \mathcal{N}(\boldsymbol 0, \boldsymbol I)$. It is then classical that the limiting eigenvalue distribution of $S$ (in the almost sure weak convergence) is given by $MP_\gamma\boxtimes\nu$, the free multiplicative convolution of the Marchenko-Pastur distribution $MP_\gamma$ of paramter $\gamma$ with the limiting eigenvalue distribution $\nu$ of $C$. Thus, you indeed obtain $MP_\gamma$ if and only if $\nu=\delta_1$, namely if and only if $$ \frac{1}{N}\sum_{i=1}^N\delta_{v_i/\sqrt{M}} \longrightarrow \delta_0\qquad \mbox{weakly as $N\rightarrow\infty$ }. $$ For a reference, you can look at the book of Anderson, Guionnet and Zeitouni, "Introduction to random matrices" (Chapter 5), but also any introduction to free probability I guess.

As you can see, free probability provides a powerful language to describe such perturbed random matrix models!

You can rewrite $$ S=\frac{1}{M}C^{1/2}XX^T(C^{1/2})^T $$ where $X_{.j}\sim \mathcal{N}(\boldsymbol 0, \boldsymbol I)$. It is then classical that the limiting eigenvalue distribution of $S$ (in the almost sure weak convergence) is given by $MP_\gamma\boxtimes\nu$, the free multiplicative convolution of the Marchenko-Pastur distribution $MP_\gamma$ of paramter $\gamma$ with the limiting eigenvalue distribution $\nu$ of $C$. Thus, you indeed obtain $MP_\gamma$ if and only if $\nu=\delta_1$, namely if and only if $$ \frac{1}{N}\sum_{i=1}^N\delta_{v_i} \longrightarrow \delta_0\qquad \mbox{weakly as $N\rightarrow\infty$ }. $$ For a reference, you can look at the book of Anderson, Guionnet and Zeitouni, "Introduction to random matrices" (Chapter 5), but also any introduction to free probability I guess.

As you can see, free probability provides a powerful language to describe such perturbed random matrix models!