a two letters alphabet, and then we indicate the way to adapt the proof for the case of $q$ letters.

Random Lyndon words are, in some sense, conditioned random words. In section \[words\], we obtain the number of Lyndon words with length $n$ by dividing the number of primitive words with length $n$ by $n$, (the shepherd’s principle: counting the legs and dividing by four to obtain the number of sheeps). Thus the typical statistical behavior of a random word and of a random Lyndon word can be easily linked (see lemma \[usarg\]).

Our analysis starts in section \[run\]: we recall, among a number of well known properties of random words with length $n$, those useful for our purposes. In particular, we study the number of runs and the length of the longest run of “$a$”, which is typically of order $\log_2 n$.

The key step is to prove that the two longest runs of “$a$” are approximately located along the word as two independent uniform random variables, and thus the distance $D_{n}$ between the first longest one and the second longest one follows approximately the uniform law on $[0..n]$. The distance $D_{n}$ is of course closely related to the 