w,U_{j_0(w)}\right) \right\|_{{\mathbb Q}\otimes {\bar{\mathbb P}_n}}^2\nonumber \\
&=&{\mathbb E}_{{\mathbb Q}\otimes {\bar{\mathbb P}_n}} \left[ d_n(\pi_x.w)  - {\mathbb E}_{{\mathbb Q}\otimes
{\bar{\mathbb P}_n}} \left( \left. d_n(\pi_x.w) \right| w,U_{j_0(w)}\right)
\right]^2 \nonumber \\
& = & {\mathbb E}_{{\mathbb Q}\otimes {\bar{\mathbb P}_n}} \left[  
\sum_{j=1}^{K_n(w)-1} \frac{|Y_j(w)|}n \mathbf{1}_{
\{U_{j}<U_{j_0(w)}\}} - \sum_{\scriptsize
\begin{array}{cc}
1 \le j \le {K_n(w)-1} \nonumber \\
j \neq j_0(w)
\end{array}}
\frac{|Y_j(w)|}n U_{j_0(w)} 
  \right]^2 \nonumber \\
& = & {\mathbb E}_{{\mathbb Q}\otimes {\bar{\mathbb P}_n}} \left[  
\sum_{\scriptsize
\begin{array}{cc}
1 \le j \le {K_n(w)-1} \nonumber \\
j \neq j_0(w)
\end{array}}
\frac{|Y_j(w)|}n \left( \mathbf{1}_{\{
U_{j}<U_{j_0(w)}\}} -U_{j_0(w)} \right)
 \right]^2 \nonumber \\
& \leq &{\mathbb E}_{{\mathbb Q}\otimes {\bar{\mathbb P}_n}} \left[  \frac14 \sum_{\scriptsize
\begin{array}{cc}
1 \le j \le {K_n(w)-1}\\
j \neq j_0(w)
\end{array}}
\left( \frac{|Y_j(w)|}{n} \right)^2
\right] \label{ccc}\\
& \leq & \frac{ 7\log_2n}{4n}, \label{ddd}\end{aligned}$$ which tends to $0$ when $n$ goes to infinity. To obtain inequality (\[ccc\]), we conditioned first on $w$ and $U_{j_0(w)}$; for inequality (\[ddd\]), we used the facts that, on $\mathcal G_n$, all blocks have length smaller than $7\log_2n$ and that $\sum_i |Y_i|=n$.

Consequently, $$\begin{aligned}
\left\| d_n o W  - U_{j_0} \right\|_{{\mathbb Q}\otimes {\bar{\mathbb P}_n}} &
\leq & \left\| d_n(\pi_x.w)  -{\mathbb E}_{{\mathbb Q}\otimes {\bar{\mathbb P}_n}} \left( \left.
d_n(\pi_x.w) \right| w,U_{j_0}(w)\right)
\right\|_{{\mathbb Q}\otimes {\bar{\mathbb P}_n}}\\
&&+ \left\| {\mathbb E}_{{\mathbb Q}\otimes {\bar{\mathbb 