  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.
