&  \quad + \left|\left(\Sigma_{ii}B^{\left(i\right)}\right) \right|  \left| \left(\Sigma_{jj}B^{\left(j\right)}\right) \right| \Bigg]\Bigg]\\
    &< \dfrac{a}{am - 1}\sum_{k=1}^{m}\left[\dfrac{b}{n^{2}}\left[\left(D\psi\left(n\right)\right)^{2} + \Sigma_{ii}\left(1 + \epsilon\right)\sqrt{2n\log\log n}\left(D\psi\left(n\right)\right)  \right. \right.\\
    & \quad \left. + \Sigma_{jj}\left(1 + \epsilon\right)\sqrt{2n\log\log n}\left(D\psi\left(n\right)\right) + \Sigma_{ii}\Sigma_{jj}\left(1 + \epsilon\right)^{2}\left(2n\log\log n\right) \right] \Bigg]\\
    &= \dfrac{am}{am - 1} \left[\dfrac{b}{n}\left(\dfrac{D \psi\left(n\right)}{\sqrt{n}}\right)^{2} + \left(\Sigma_{ii} + \Sigma_{jj}\right)\left(1 + \epsilon\right)\dfrac{b \sqrt{2\log\log n}}{n}\left(\dfrac{D\psi\left(n\right)}{\sqrt{n}}\right) \right.\\
    & \quad + \left. 2\Sigma_{ii}\Sigma_{jj}\left(1 + \epsilon\right)^{2}\left(\dfrac{b\log\log n}{n}\right) \right]\\
     &= \dfrac{am}{am - 1} \dfrac{b \log \log n}{n}\left[\left(\dfrac{D \psi\left(n\right)}{\sqrt{n \log \log n}}\right)^{2} + \sqrt{2} \left(\Sigma_{ii} + \Sigma_{jj}\right)\left(1 + \epsilon\right)\left(\dfrac{D\psi\left(n\right)}{\sqrt{n \log \log n}}\right) +  2\Sigma_{ii}\Sigma_{jj}\left(1 + \epsilon\right)^{2}\right]\\
    & \xrightarrow{a.s} 0 \; \; n \to \infty \, .\end{aligned}$$ Following the same steps for $B^{ij}$ with $n$ replaced by $mn$, $\left|B^{ij} \right| \xrightarrow{a.s} 0 \; \; n \to \infty$. Therefore $\|A\| \xrightarrow{a.s} \mathbf{0}$ and $\|B\| \xrightarrow{a.s} \mathbf{0}$ as $n \to \infty$. Using this, $$\label{eq:Y_diff_term_0}
  \left|\left|\dfrac{ab}{am - 1}\sum_{k=1}^{m}\left(\hat{\mu}_k-\hat{\mu}\right)\left(\hat{\mu}_k-\hat{\mu}\right)^{T}\right|\right| = \left|\left|A - B \right|\right| \leq \|A\| + \|B\| \xrightarrow{a.s} \mathbf{0} \; \; n \to \infty \,.$$

Using and consistency of $\hat{\Sigma}_{k,b}$ for all $k$ in $$\begin{aligned}
  \left|\left|\hat{\Sigma}_{R,b} - \Sigma   \right|\right| & =  \left|\left|\dfrac{m(a-1)}{am-1} \hat{\Sigma}_{A,b} - \Sigma +   \dfrac{ab}{am-1}\sum_{k=1}^{m}\left(\hat{\mu}_k-\hat{\mu}\right)\left(\hat{\mu}_k-\hat{\mu}\right)^{T}\right|\right| \\
  & \leq \left|\left| \dfrac{m(a-1)}{am-1} \hat{\Sigma}_{A,b} - \Sigma \right|\right| +  \left|\left|  \dfrac{ab}{am-1}\sum_{k=1}^{m}\left(\hat{\mu}_k-\hat{\mu}\right)\left(\hat{\mu}_k-\hat{\mu}\right)^{T}  \right|\right|\\
  & \overset{a.s.}{\to} 0 \quad \text{ as } n \to \infty\,.\end{aligned}$$

Bias of RBM {#subsec:bias_RBM}
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