large deviations are well known (see the book by Bollob[á]{}s [@Bollobas Th. 7, p.13] for instance): there exists a positive constant $C'_\gamma$ such that $$\begin{aligned}
{\mathbb P_n}\left( \left| N_n-\frac{n}2 \right| \ge \gamma n^{-1/2+\varepsilon}
\right)
& \le &  O \left( \exp \left( -C_\gamma' n^{2\varepsilon}
  \right) \right).\end{aligned}$$

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 $N_n(w) -1 \leq N_n(\rho(w))
\leq N_n(w)$. Thus, using Lemma \[usarg\], we obtain the announced result.

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The next step is to study the length of the longest run of a word $w \in \mathcal A^n$. For this mean, we will use the following construction of the uniform probability measure on the set of all infinite words on $\mathcal A$:

\[cons2\] Let $(Z_i)_{i \in {\mathbb N}}$ be independent identically distributed geometrical random variables with parameter $1/2$ defined on a probability space $(\Omega, \mathcal F, {\mathbb P})$ and let $\epsilon$ be a Bernoulli random variable with parameter $1/2$ defined on $\Omega$ and independent of the $Z_i$’s. To obtain a infinite random sequence of $a$ and $b$, do the following:

-   if $\epsilon=1$, write $Z_1$ “$a$”, followed 