Limit law of the standard right factor of a random Lyndon word
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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: $$μ(dx)=\frac1q δ_{1}(dx) + \frac{q-1}q \mathbf{1}_{[0,1)}(x)dx,$$ 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.