| Issue |
RAIRO-Theor. Inf. Appl.
Volume 58, 2024
Randomness and Combinatorics - Edited by Luca Ferrari & Paolo Massazza
|
|
|---|---|---|
| Article Number | 11 | |
| Number of page(s) | 14 | |
| DOI | https://doi.org/10.1051/ita/2024005 | |
| Published online | 26 March 2024 | |
On a probabilistic extension of the Oldenburger–Kolakoski sequence
1
École Normale Supérieure de Lyon, 15 parvis René Descartes, F-69342 Lyon, France
2
Université de Lorraine, Loria, UMR 7503, Vandœuvre-lès-Nancy F-54506, France
3
Univ Rouen Normandie, CNRS, Normandie Univ, LMRS UMR 6085, F-76000 Rouen, France
* Corresponding author: damien.jamet@loria.fr
Received:
14
December
2022
Accepted:
23
February
2024
The Oldenburger–Kolakoski sequence is the only infinite sequence over the alphabet {1, 2} that starts with 1 and is its own run-length encoding. In the present work, we take a step back from this largely known and studied sequence by introducing some randomness in the choice of the letters written. This enables us to provide some results on the convergence of the density of 1’s in the resulting sequence. When the choice of the letters is given by an infinite sequence of i.i.d. random variables or by a Markov chain, the average densities of letters converge. Moreover, in the case of i.i.d. random variables, we are able to prove that the densities even almost surely converge.
Mathematics Subject Classification: 11K31 / 11K36
Key words: Kolakoski–Oldenburger / combinatorics on words / random variable / Markov chain
© The authors. Published by EDP Sciences, 2024
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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