the words begins with “$a$” or “$b$”. Taking $y= (1 -\varepsilon)\log_2n$, we obtain easily: $$\label{minmax} {\mathbb P_n}\left( M_n^a \leq (1 -\varepsilon) \log_2n
\right) \leq O \left( \exp \left( -\frac{n^\varepsilon}4 \right)
\right).$$

To obtain the same inequality for the probability measure ${\tilde{\mathbb P}_n}$ on the set $\mathcal L_n$ of Lyndon words with length $n$, note that, for a primitive word $w$, we have $M_n^a( w) \leq
 M_n^a(\rho(w))$. We can now use lemma \[usarg\] to conclude.

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\[lemMnGV\] For any $1<A \le 2$, $${\tilde{\mathbb P}_n}\left( M_n^a \ge A\log_2{n} \right) \le {\tilde{\mathbb P}_n}\left( M_n \ge A\log_2{n}
\right)
 \leq   O \left( n^{1-A} \right).$$

**Proof.** As before, we begin by proving the above inequality for the probability measure ${\mathbb P_n}$ on the set $\mathcal A^n$. Note that we still have $M_n\ge M_n^a$. Thus for $y>0$, we have: $$\begin{aligned}
{\mathbb P_n}(M_n^a \leq y )  
 & \geq & {\mathbb P_n}(M_n \leq y ) \nonumber \\
 & = & {\mathbb P_n}( \forall i \in \{1,
\dots,N_n\}, \; X_i \leq
y) \nonumber \\
& \geq & {\mathbb P}(\forall i \in \{1, \dots,N_n\}, \; Z_i \leq y) \label{aaa}\\
& \geq & {\mathbb P}(\forall i \in \{1, \dots,n\}, \; Z_i \leq y) \label{bbb}\\
 & \geq &  (1-2^{- \lfloor y \rfloor})^n.\nonumber \end{aligned}$$ 