Lyndon word are essential for the analysis of the building algorithm.

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For any Lyndon word $w \in \mathcal L_n$, let $R_{n}(w)$ denote the length of its standard right factor. Endowing $\mathcal L_n$ with the uniform probability measure makes $R_n$ a random variable on $\mathcal L_n$. Bassino, Cl[é]{}ment and Nicaud [@Bassino], with the help of generating functions, prove that the expectation ${\mathbb E}(R_n)$, in the case of a two letters alphabet, is asymptotically equal to $3n/4$. The aim of this paper is to determine the limit distribution of $R_n/n$ as $n$ goes to infinity:

\[loilimiteq\] For a totally ordered alphabet with $q$ letters, the normalized length $R_n/n$ of a random Lyndon word of length $n$ converges in distribution, when $n$ goes to infinity, to $$\mu(dx)= \frac1q \delta_{1}(dx) + \frac{q-1}q \mathbf{1}_{[0,1)}(x)dx,$$ where $\delta_1$ denotes the Dirac mass on the point $1$, and $dx$ Lebesgue’s measure on ${\mathbb R}$. All the moments of $R_n/n$ also converge to the corresponding moments of the limit distribution.

**Remark.** In the case $q=2$, this result was conjectured by Bassino, Cl[é]{}ment and Nicaud [@Bassino]. Simulations were also provided in this paper.

For the proof of this result, we focus first on the case of 