n^{\varepsilon} \right) \right).\end{aligned}$$ This proves the lemma for random words.

For random Lyndon words, note that if $w$ is a primitive word, then $H_n(w) \leq H_n(\rho(w)) \leq H_n(w)+1$ and use lemma \[usarg\].

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The length of the long blocks has been chosen large enough to ensure that two long blocks are distinct with high probability:

\[En\] Denote by $E_n$ the event that a word with length $n$ has at least two equal disjoint subwords with length at least $3\log_2n$. Then: $${\tilde{\mathbb P}_n}(E_n) \leq
O \left(n^{-1} \right).$$

**Proof.** We begin as usual with random words. By counting the number of possible subwords with length $3\log_2n$ and their possible positions, we have: $${\mathbb P_n}\left(E_n \right)
\leq O \left\{ n^2 (2^{3{\log_2n}})(2^{-3{\log_2 n}})^2
\right\}\leq O \left(n^{-1}\right).$$ Lemma \[usarg\] gives the same estimate for Lyndon words.

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We also want that the long blocks do not overlap with high probability, or, in other words, that the beginnings of long blocks are far away enough with high probability. This is ensured by the next lemma:

\[Dn\] Let $D_n$ be the event that there exist at least two long blocks which begin at a distance less than $8\log_2n$. Then: $${\tilde{\mathbb P}_n}(D_n) \leq
O(n^{-(1-2\varepsilon)}\log_2 n ).$$

**Proof.** As 