On a probabilistic extension of the Oldenburger-Kolakoski sequence

TL;DR

This paper introduces a probabilistic extension of the Oldenburger-Kolakoski sequence by incorporating randomness in letter choice, and studies the convergence of letter densities under stochastic models like i.i.d. sequences and Markov chains.

Contribution

It extends the classical sequence by adding probabilistic elements and analyzes the convergence properties of letter densities in this new stochastic setting.

Findings

Letter densities converge when choices are modeled by i.i.d. sequences or Markov chains.

Almost sure convergence of densities is proved for i.i.d. random choices.

The study provides new insights into the probabilistic behavior of a classical combinatorial sequence.

Abstract

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.