usual, we start with the case of random words: $$\begin{aligned}
{\mathbb P_n}(D_n) & \leq & {\mathbb P_n}\left( H_n \geq \frac94 n^\varepsilon
\right)
+ {\mathbb P_n}\left(D_n\cap \left\{ H_n<
\frac 94 n^\varepsilon\right\} 
\right) \end{aligned}$$ Let us denote by $F_n$ the last event. On $F_n$, at least one of the $H_n$ subwords with length $8\log_2
n$ starting just after a run of “$a$” with length at least $(1-\varepsilon)\log_2 n$ must admit a subword of “$a$” with length $(1-\varepsilon)\log_2 n$ (which is the beginning of the next long block). By an estimate analogous to the one used in the previous lemma, $${\mathbb P_n}\left(F_n \right)
\leq O \left\{ \frac 94 n^\varepsilon 8\log_2n
2^{-(1-\varepsilon){\log_2 n}}
\right\}=O( n^{-(1-2\varepsilon)} \log_2 n ).$$ By lemma \[lemHn\], the first term is negligible, and Lemma \[usarg\] concludes for Lyndon words.

Now we consider the set of “good” Lyndon words that satisfy all the previous properties:

Denote by $\mathcal G_n$ the set of Lyndon words $w$ satisfying the following conditions:

-   the maximal run of “$a$” satisfies $(1-\varepsilon)
    \log_2{n}  \leq M_n^a \leq 2 \log_2{n}$

-   the maximal run satisfies $(1-\varepsilon) \log_2{n}  \leq M_n \leq 2 \log_2{n}$

-   the number of long blocks satisfies $\displaystyle \frac14
    n^\varepsilon \leq H_n \leq \frac94 n^\varepsilon  $;

