---
abstract: 'Consider the set of finite words on a totally ordered alphabet with $q$ letters. We prove that the distribution of the length of the standard right factor of a random Lyndon word with length $n$, divided by $n$, converges to: $$\mu(dx)=\frac1q \delta_{1}(dx) + \frac{q-1}q \mathbf{1}_{[0,1)}(x)dx,$$ when $n$ goes to infinity. The convergence of all moments follows. This paper completes thus the results of [@Bassino], giving the asymptotics of the mean length of the standard right factor of a random Lyndon word with length $n$ in the case of a two letters alphabet.'
address: |
    Institut Elie Cartan Nancy (math[é]{}matiques)\
    Universit[é]{} Henri Poincar[é]{} Nancy 1\
    Campus Scientifique, BP 239\
    54506 Vandoeuvre-l[è]{}s-Nancy Cedex France
author:
- 'R[é]{}gine Marchand'
- Elahe Zohoorian Azad
bibliography:
- 'Lyndon.bib'
title: Limit law of the length of the standard right factor of a Lyndon word
---

Introduction {#intro}
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Consider a finite totally ordered alphabet $\mathcal A$ and for each $n \in
\mathbb{N}=\{1,2,3,\dots\}$, denote by $\mathcal A^n$ the set of words with length $n$ on $\mathcal A$. A Lyndon word with length $n$ is a word in $\mathcal A^n$ which is strictly smaller, for the lexicographic order, than each of its proper suffixes. We denote 