Limit law of the standard right factor of a random Lyndon word

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 \cite{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.