be found by the inversion formula $$\label{eq:inver}
G([a,b])=\frac{1}{\pi}\lim_{\epsilon\rightarrow
0^+}\int_{a}^b\mathrm{Im} \left(m_G(s+i\epsilon)\right)ds.$$ The properties of the Stieltjes transform of $F_N(x)$ has been studied by Bai, Silverstein and Choi in the papers [@CS], [@BS95], [@BS98], [@BS99], [@S95]. We will now summarize the results that we need from these papers.

First let us denote the e.d.f of the eigenvalues of $\Sigma_N$ by $H_N$, that is, we have $$dH_N(x)=\frac{1}{N}\sum_{j=1}^N\delta_{a_j}.$$ Furthermore, we assume that as $N\rightarrow\infty$, the distribution $H_N$ weakly converges to a distribution function $H$. Then as $N\rightarrow\infty$, the e.d.f $F_N(x)$ converges weakly to a nonrandom p.d.f $F$, and that the Stieltjes transform $m_F$ of $F(x)$ satisfies the following equation [@CS], [@S95] $$\label{eq:algeq}
m_F(z)=\int_{\mathbb{R}}\frac{1}{t(1-c-czm_F)-z}dH(t).$$ Let us now consider the closely related matrix $\underline{B}_N$ $$\label{eq:BNline}
\underline{B}_N=\frac{1}{N}X\Sigma_NX^{\dag}.$$ The matrix $\underline{B}_N$ has the same eigenvalues as $B_N$ together with an addition $M-N$ zero eigenvalues. Therefore the e.d.f $\underline{F}_N$ of the eigenvalues of $\underline{B}_N$ are related to $F_N$ by $$\label{eq:FNline}
\underline{F}_N=(1-c_N)I_{[0,\infty)}+c_NF_N,\quad c_N=\frac{N}{M}.$$ where $I_{[0,\infty)}$ is the step function that is $0$ on $\mathbb{R}_-$ and $1$ on $\mathbb{R}_+$. In particular, as $N\rightarrow\infty$, the distribution $\underline{F}_N$ converges weakly to a p.d.f $\underline{F}$ that is related to $F$ by $$\label{eq:Fline}
\underline{F}=(1-c)I_{[0,\infty)}+cF$$ and their Stieltjes transforms are related by $$\label{eq:stielim}
m_{\underline{F}}(z)=-\frac{1-c}{z}+cm_F(z).$$ Then from 