\hat{\mu}\right)\left(\bar{Y}_{k,s} - \hat{\mu}_k + \hat{\mu}_k - \hat{\mu}\right)^{T}\\
&= \dfrac{b}{am-1}\sum_{k=1}^{m}\sum_{s=1}^{a}\left[\left(\bar{Y}_{k,s}-\hat{\mu}_k\right)\left(\bar{Y}_{k,s}-\hat{\mu}_k\right)^{T} + \left(\bar{Y}_{k,s}-\hat{\mu}_k \right)\left(\hat{\mu}_k - \hat{\mu}\right)^{T}  \right.\\
& \quad \quad  \left. + \left(\hat{\mu}_k-\hat{\mu}\right)\left(\bar{Y}_{k,s}-\hat{\mu}_k\right)^{T} + \left(\hat{\mu}_k-\hat{\mu}\right)\left(\hat{\mu}_k-\hat{\mu}\right)^{T}\right]\\
&= \dfrac{b}{am-1}\dfrac{a-1}{b}\sum_{k=1}^{m}\hat{\Sigma}_{k,b} + \dfrac{b}{am-1}\sum_{k=1}^{m}\left[\sum_{s=1}^{a}\left(\bar{Y}_{k,s}-\hat{\mu}_k\right)\right]\left(\hat{\mu}_k-\hat{\mu}\right)^{T} \\
& \quad \quad + \dfrac{b}{am-1}\sum_{k=1}^{m}\left(\hat{\mu}_k-\hat{\mu}\right)\sum_{s=1}^{a}\left(\bar{Y}_{k,s}-\hat{\mu}_k\right)^{T} + \dfrac{ab}{am-1}\sum_{k=1}^{m}\left(\hat{\mu}_k-\hat{\mu}\right)\left(\hat{\mu}_k-\hat{\mu}\right)^{T}\\
&= \dfrac{m(a-1)}{am-1} \hat{\Sigma}_{A,b} + \dfrac{ab}{am-1}\sum_{k=1}^{m}\left(\hat{\mu}_k-\hat{\mu}\right)\left(\hat{\mu}_k-\hat{\mu}\right)^{T} \,. {\addtocounter{equation}{1}\tag{\theequation}}\label{eq:repli_breakdown}\end{aligned}$$

By Lemma \[lem:Y\_diff\_simpli\] $$\begin{aligned}
\dfrac{ab}{am - 1}\sum_{k=1}^{m}\left(\hat{\mu}_k-\hat{\mu}\right)\left(\hat{\mu}_k-\hat{\mu}\right)^{T} 
&= \dfrac{ab}{am - 1}\sum_{k=1}^{m}\left(\hat{\mu}_k-\mu\right)\left(\hat{\mu}_k-\mu\right)^{T} - \dfrac{abm}{am - 1}\left(\hat{\mu}-\mu\right)\left(\hat{\mu}-\mu\right)^{T}\end{aligned}$$

Let $$A = \dfrac{a}{am - 1}\sum_{k=1}^{m}b\left(\hat{\mu}_k-\mu\right)\left(\hat{\mu}_k-\mu\right)^{T} \quad \text{ and }  \quad B = \dfrac{abm}{am - 1}\left(\hat{\mu}-\mu\right)\left(\hat{\mu}-\mu\right)^{T}\,.$$ By [@stra:1964], $\psi(n) = O(\sqrt{n \log \log n})$ so that the rate is such that the law of iterated logarithm holds. Let $A^{ij}$, be the $(i,j)$th element of the matrix $A$. Using the SIP and Lemma \[lemma: brownian\_bound\], $$\begin{aligned}
    \left |A^{ij} \right | &= \dfrac{a}{am - 1}\sum_{k=1}^{m} \left| b\left(\bar{Y}^{\left(i\right)}_{k} - \mu^{(i)}\right)\left(\bar{Y}^{\left(j\right)}_{k} - \mu^{(j)}\right) \right|\\
    & \leq \dfrac{a}{am - 1}\sum_{k=1}^{m}\left[ \left| \dfrac{b}{n^{2}}\left(\sum_{r=1}^{n}Y^{\left(i\right)}_{r,k} - n\mu^{(i)}\right)\left(\sum_{s=1}^{n}Y^{\left(j\right)}_{s,k} - n\mu^{(j)}\right) \right|  \right]\\
    &= \dfrac{a}{am - 1}\sum_{k=1}^{m}\left[\dfrac{b}{n^{2}}  \left|\left(\sum_{r=1}^{n}Y^{\left(i\right)}_{r,k} - n\mu^{(i)} - \Sigma_{ii}B^{\left(i\right)} + \Sigma_{ii}B^{\left(i\right)}\right)\left(\sum_{s=1}^{n}Y^{\left(j\right)}_{s,k} - n\mu^{(j)} - \Sigma_{jj}B^{\left(j\right)} + \Sigma_{jj}B^{\left(j\right)}\right) \right| \right]\\
    & \leq \dfrac{a}{am - 1}\sum_{k=1}^{m}\left[\dfrac{b}{n^{2}}\left[ \left| \left(\sum_{r=1}^{n}Y^{\left(i\right)}_{r,k} - n\mu^{(i)} - \Sigma_{ii}B^{\left(i\right)}\right)\left(\sum_{s=1}^{n}Y^{\left(j\right)}_{s,k} - n\mu^{(j)} - \Sigma_{jj}B^{\left(j\right)}\right) \right| \right. \right.\\
    & \left. \left. \quad + \left|\left(\Sigma_{ii}B^{\left(i\right)}\right) \right|    \left|\left(\sum_{s=1}^{n}Y^{\left(j\right)}_{s,k} - n\mu^{\left(j\right)} - \Sigma_{jj}B^{\left(j\right)}\right)\right|   +   \left|\left(\Sigma_{jj}B^{\left(j\right)}\right) \right|   \left|\left(\sum_{r=1}^{n}Y^{\left(i\right)}_{r,k} - n\mu^{(i)} - \Sigma_{ii}B^{\left(i\right)}\right) \right| \right. \right.\\
    