of notation, set $C = m\left(a-1\right)/\left(am - 1\right)$. Further, define $$\begin{aligned}
g_1\left(n\right) &= 2\left(\dfrac{1}{1-c} \dfrac{am}{am - 1} +  \dfrac{c}{1-c} \dfrac{am}{arm - 1} \right)\dfrac{b \psi\left(n\right)^2}{n^2} \\ 
g_2\left(n\right) &= 2\left(\dfrac{1}{1-c} \dfrac{am}{am - 1} +  \dfrac{c}{1-c} \dfrac{am}{arm - 1} \right) \left(\Sigma_{ii}+\Sigma_{jj}\right)\left(1+\epsilon\right)\dfrac{b\psi\left(n\right) \sqrt{2n \log\log n}}{n^2}\\
g_3\left(n\right) &= 2\left(\dfrac{1}{1-c} \dfrac{am}{am - 1} +  \dfrac{c}{1-c} \dfrac{am}{arm - 1} \right)\Sigma_{ii}\Sigma_{jj}\left(1 + \epsilon\right)^2 \dfrac{2b \log\log n}{n}\,.\end{aligned}$$ Since $\psi(n) = O(\sqrt{n \log \log n})$, if $\left(b\log \log n\right)/n \to 0$, then $g_1, g_2, g_3 \to 0$. From the steps in Theorem \[thm:RBM\_consistency\], $$\begin{aligned}
 &\left| \hat{\Sigma}_{R,L}^{ij} - C \hat{\Sigma}_{A,L}^{ij} \right|\\
 & =  \left| \dfrac{1}{1-c} \left(\hat{\Sigma}_{R,b}^{ij} - C \hat{\Sigma}_{A,b}^{ij} \right) - \dfrac{c}{1-c} \left(\hat{\Sigma}_{R,b/r}^{ij} - C \hat{\Sigma}_{A,b/r}^{ij} \right)  \right|\\ 
 & = \left| \dfrac{1}{1-c} \left( \dfrac{ab}{am - 1} \sum_{k=1}^{m} \left(\hat{\mu}_{k}^{(i)} - \hat{\mu}^{(i)}\right)\left(\hat{\mu}_{k}^{(j)} - \hat{\mu}^{(j)}\right)^T \right) - \dfrac{c}{1-c} \left( \dfrac{ab}{arm - 1} \sum_{k=1}^{m} \left(\hat{\mu}_{k}^{(i)} - \hat{\mu}^{(i)}\right)\left(\hat{\mu}_{k}^{(j)} - \hat{\mu}^{(j)}\right)^T  \right) \right| \\
 &\leq \dfrac{1}{1-c} \dfrac{ab}{am - 1}\left| \sum_{k=1}^{m} \left(\hat{\mu}_{k}^{(i)} - \hat{\mu}^{(i)}\right)\left(\hat{\mu}_{k}^{(j)} - \hat{\mu}^{(j)}\right)^T \right| + \dfrac{c}{1-c} \dfrac{ab}{arm - 1}\left| \sum_{k=1}^{m} \left(\hat{\mu}_{k}^{(i)} - \hat{\mu}^{(i)}\right)\left(\hat{\mu}_{k}^{(j)} - \hat{\mu}^{(j)}\right)^T \right| \\ 
 & =  \left(\dfrac{1}{1-c} \dfrac{ab}{am - 1} +  \dfrac{c}{1-c} \dfrac{ab}{arm - 1} \right)\left| \sum_{k=1}^{m} \left(\hat{\mu}_{k}^{(i)} - \hat{\mu}^{(i)}\right)\left(\hat{\mu}_{k}^{(j)} - \hat{\mu}^{(j)}\right)^T \right|\\
 & \leq D^2g_1\left(n\right) + Dg_2\left(n\right) + g_3\left(n\right)\,.\end{aligned}$$ By , there exists an $N_0$ such that $$\begin{aligned}
\left(\hat{\Sigma}_{R,L}^{ij} - C 