long enough to be distinct with high probability. We study then the positions of these long blocks along the word $w$. Here is our definition:

Let $w$ be a word with length $n$. The long blocks of $w$ are the subwords of $w$ that:

-   begin with a run of “a” with length equal or greater than $(1-\varepsilon)\log_2 n$,

-   end with a run of “b”

-   have the smallest possible length larger than $3\log_2n$.

We denote by $H_n$ the number of long blocks.

The next lemma estimates the number of long blocks for a random Lyndon word. Note that although the crude estimate we give could be sharpen, it is sufficient for our mean.

\[lemHn\] There exists a constant $D>0$ such that $$\begin{aligned}
{\tilde{\mathbb P}_n}\left( \frac14  n^\varepsilon \leq H_n \leq \frac94
n^\varepsilon
\right) & \ge & 1-O \left( \exp \left(-D
    n^{\varepsilon} \right) \right).\end{aligned}$$

**Proof.** We begin once again by proving the inequality for the probability measure ${\mathbb P_n}$ on the set $\mathcal A^n$. Set, for $i \geq 1$, $B_{i} =
\mathbf{1}_{\{ Z_{i}\geq(1-\varepsilon)\log_{2}{n}\}}.$ Then $(B_i)_{i \geq 1}$ are independent identically distributed Bernoulli random variables with parameter $p_{n, \varepsilon}$, which satisfies $n^{\varepsilon-1} \leq p_{n, \varepsilon} \leq
2n^{\varepsilon-1}.$ Note that $$\begin{aligned}
\sum_{1 