G_n \\
(x, w) & \longmapsto & \pi_x.w
\end{array}
\right.$$ follows the uniform law on $\mathcal G_n$. We can now study, under the uniform probability ${\bar{\mathbb P}_n}$ on $\mathcal G_n$, the position of the second smallest block defined by $$d_n(w)=\frac1n \sum_{i=0}^{j_0(w)-1} |Y_i(w)|.$$ Here $|w|$ denotes the length of the word $w$. Thus, the random variable $$\begin{array}{rrll}
d_n o W: & X \times  \mathcal G_n  &
\longrightarrow &
[0,1] \\
& (x, w) & \longmapsto & d_n(\pi_x.w)
\end{array}$$ has the same law as the random variable $d_n$ under ${\bar{\mathbb P}_n}$.

We will thus focus on this new random variable to use the property of invariance under the permutation of blocks. Remember that the convergence in $L^2$ implies the convergence in probability; thus, in the following, the notation $\|.\|_{{\mathbb P}}$ will denotes the $L^2$-norm associated to a probability measure ${\mathbb P}$.

\[distance\] We have: $$\|d_no W-U_{j_0}\|_{{\mathbb Q}\otimes {\bar{\mathbb P}_n}} \le
O \left( \left[ \frac{\log_2n}{n} \right]^{1/2}\right).$$ This implies in particular that the law of $d_n$ under ${\bar{\mathbb P}_n}$ converges weakly to the uniform law on $[0,1]$ and that every moment of $d_n$ converges to the corresponding moment of the uniform distribution.

**Remark.** Coming back to random words, this result implies that the normalized 