} \left( \mathbb{E}\left[ \chi_{n,i}^{\geq k}  \right] \mathbb{E}\left[ \chi_{n,j}^{\geq k}  \right] + \mathbb{P}(C_{ij}^c) \right),
\end{align*}
which in turn implies that
\begin{align*}
\frac{1}{n^2} \sum_{i=1}^n \sum_{j \neq i} \mathbb{E}\left[ \left( \chi_{n,i}^{\geq k} -  \mathbb{E}\left[ \chi_{n,i}^{\geq k} \right]  \right) \left( \chi_{n,j}^{\geq k} -  \mathbb{E}\left[ \chi_{n,j}^{\geq k} \right]  \right) \right] \leq \frac{1}{n^2} \sum_{i=1}^n \sum_{j \neq i} \mathbb{P}(C_{ij}^c). 
\end{align*}

Similarly to what was done on the graph, define $ \hat{\mathcal{N}}_i^{(k)} = \hat I_{\hat\sigma_{k,i}^-}^-  \cup  \hat I_{\hat \sigma_{k,i}^+}^+  \cup \hat A_{\hat \sigma_{k,i}^+}^+ \cup \hat A_{\hat \sigma_{k,i}^-}^-$ to be the set of {\em identities} that appear during the construction of the double tree $(\mathcal{T}_\mu^+(\kappa_m;\bold{X}_i), \mathcal{T}_\mu^-(\kappa_m;\bold{X}_i))$ up to Step $\hat{\sigma}_{k,i}^+$ on the inbound side, and up to Step $\hat{\sigma}_{k,i}^-$ on the outbound side. Let $\hat{C}_{ij}= \left\{ \hat{\mathcal{N}}_i^{(k)}\cap\hat{\mathcal{N}}_j^{(k)}=\varnothing \right\}$. We then have
\begin{align*}
\mathbb{P}(C_{ij}^c) &\leq \mathbb{P}(C_{ij}^c,\tau_i^+>\sigma_{k,i}^+,\tau_i^->\sigma_{k,i}^-,\tau_j^+>\sigma_{k,j}^+,\tau_j^->\sigma_{k,j}^-) 1(\Omega_{m,n})  + 1(\Omega_{m,n}^c) \\
&\hspace{5mm} + \mathbb{P}( \{ \tau_i^+ \leq \sigma_{k,i}^+ \} \cup \{\tau_i^-\leq \sigma_{k,i}^-\} )+\mathbb{P}( \{ \tau_j^+ \leq \sigma_{k,j}^+ \} \cup \{ \tau_j^- \leq \sigma_{k,j}^-\} )  \\
&\leq 1(\Omega_{m,n}^c) +  \mathbb{P}(\hat{C}_{ij}^c, \, |\hat{\mathcal{N}}_i^{(k) }  | \leq \log{n}) 1(\Omega_{m,n}) +\mathbb{P}(|\hat{\mathcal{N}}_i^{(k) }  | >\log{n}) \\
&\hspace{5mm} + \mathbb{P}_i( \{ \tau^+ \leq \sigma_{k}^+ \} \cup \{\tau^-\leq \sigma_{k}^-\} )+\mathbb{P}_j( \{ \tau^+ \leq \sigma_{k}^+ \} \cup \{ \tau^- \leq \sigma_{k}^-\} ),
\end{align*}
where the event $\Omega_{m,n}$ is defined in Theorem~\ref{T.Coupling}. 

To bound the first probability on the right-hand side, define $\hat{\mathcal{F}}_{k,i}$ to be the sigma-algebra generated by the construction of the double tree whose root has {\em identity} $i$, up to Step $\hat{\sigma}_{k,i}^+$ on the inbound side and up to Step $\hat{\sigma}_{k,i}^-$ on the outbound side. 
Now note that
\begin{align*}
\hat{C}_{ij} &= \{ j \notin \hat{\mathcal{N}}_i^{(k)} \} \cap \left(\bigcap_{r=1}^{\hat{\sigma}_{k,j}^-}\bigcap_{t\in \hat{\mathcal{N}}_i^{(k)}} \{\tilde{Z}_{\hat T_{r,j}^-,t}=0 \} \right) \cap \left(\bigcap_{r=1}^{\hat{\sigma}_{k,j}^+} \bigcap_{t\in \hat{\mathcal{N}}_i^{(k)}} \{ Z_{t,\hat T_{r,j}^+}=0 \} \right)
\end{align*}
where $\hat T_{r,j}^+$ and $\hat T_{r,j}^-$ are the $r$th active {\em identities} to have their offspring sampled in the double tree whose root is $j$. Moreover, if we define $B_{s} = \bigcap_{t\in \hat{\mathcal{N}}_i^{(k)}} \{ Z_{ts}=0 \}$ and $\tilde B_{s} = \bigcap_{t\in \hat{\mathcal{N}}_i^{(k)}} \{ \tilde Z_{st}=0 \}$, then
$$\{ j \notin \hat{\mathcal{N}}_i^{(k)} \} = B_{j} \cap \tilde B_{j} \qquad \text{and} \qquad \hat C_{ij} = B_{j} \cap \tilde B_{j} \cap \left( \bigcap_{r=1}^{\hat{\sigma}_{k,j}^-} \tilde B_{\hat T_{r,j}^-} \right) \cap \left( \bigcap_{r=1}^{\hat{\sigma}_{k,j}^+} B_{\hat T_{r,j}^+} \right),$$
and therefore, since $\hat \sigma_{k,j}^+, \hat \sigma_{k,j}^+ \leq k$, the union bound gives
\begin{align}
\mathbb{P}( \hat C_{ij}^c | \hat{\mathcal{F}}_i^{(k)})  &\leq   \mathbb{P}\left( \left.  B_{j}^c  \cup  \left( \bigcup_{r=1}^{\hat{\sigma}_{k,j}^+} B_{\hat T_{r,j}^+}^c\right)  \right|  \hat{\mathcal{F}}_i^{(k)} \right) +  \mathbb{P}\left( \left.   \tilde B_{j}^c \cup \left( \bigcup_{r=1}^{\hat{\sigma}_{k,j}^-} \tilde B_{\hat T_{r,j}^-}^c \right)   \right|  \hat{\mathcal{F}}_i^{(k)} \right) \notag  \\
&=  \mathbb{E}\left[ \left.  1(B_{j}^c) + \sum_{r=1}^{\hat{\sigma}_{k,j}^+}  1\left( B_j \cap \bigcap_{s=1}^{r-1} B_{\hat T_{s,j}^+} \cap B_{\hat T_{r,j}^+}^c \right)   \right|  \hat{\mathcal{F}}_i^{(k)} \right] \notag \\
&\hspace{5mm} + \mathbb{E}\left[ \left.  1(\tilde B_{j}^c) + \sum_{r=1}^{\hat{\sigma}_{k,j}^-}  1\left( \tilde B_j \cap \bigcap_{s=1}^{r-1} \tilde B_{\hat T_{s,j}^-} \cap \tilde B_{\hat T_{r,j}^-}^c \right)   \right|  \hat{\mathcal{F}}_i^{(k)} \right]  \notag \\ 
&\leq \mathbb{E}\left[ \left.  1(B_{j}^c) + \sum_{r=1}^k  1\left( \hat A_{r-1,j}^+ \neq \varnothing, \, \bigcap_{s=1}^r \{ \hat T_{s,j}^+ \notin \hat{\mathcal{N}}_i^{(k)} \},  B_{\hat T_{r,j}^+}^c \right)   \right|  \hat{\mathcal{F}}_i^{(k)} \right] \label{eq:InboundNoI} \\
&\hspace{5mm} +  \mathbb{E}\left[ \left.  1(\tilde B_{j}^c) + \sum_{r=1}^k  1\left( \hat A_{r-1,j}^- \neq \varnothing, \, \bigcap_{s=1}^r \{ \hat T_{s,j}^- \notin \hat{\mathcal{N}}_i^{(k)}\},  \tilde B_{\hat T_{r,j}^-}^c \right)   