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arXiv:1004.4062 (Published 2010-04-23)
Asymptotic behavior of some factorizations of random words
Categories: math.PRThis paper considers the normalized lengths of the factors of the Lyndon decomposition of finite random words with $n$ independent letters drawn from a finite or infinite totally ordered alphabet according to a general probability distribution. We prove, firstly, that the limit law of the lengths of the smallest Lyndon factors is a variant of the stickbreaking process. Convergence of the distribution of the lengths of the longest factors to a Poisson-Dirichlet distribution follows. Secondly, we prove that the distribution of the normalized length of the standard right factor of a random $n$-letters long Lyndon word, derived from such an alphabet, converges, when $n$ is large, to: $$ \mu(dx)=p_1 \delta_{1}(dx) + (1-p_1) \mathbf{1}_{[0,1)}(x)dx, $$ in which $p_1$ denotes the probability of the smallest letter of the alphabet.
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arXiv:math/0407016 (Published 2004-07-01)
Limit law of the standard right factor of a random Lyndon word
Categories: math.PRConsider 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.