Self-similar and Markov composition structures
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part XIII, Tome 326 (2005), pp. 59-84
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The bijection between composition structures and random closed
subsets of the unit interval implies that the composition structures
associated with $S\cap[0,1]$ for a self-similar random set
$S\subset{\mathbb R}_+$ are those which are consistent
with respect to a simple truncation operation. Using the standard coding
of compositions by finite strings of binary digits starting with a 1,
the random composition of $n$ is defined by the first $n$ terms of a random binary sequence of infinite length.
The locations of 1s in the sequence are the places visited by an increasing time-homogeneous Markov chain on the positive integers if and
only if $S=\exp(-W)$ for some stationary regenerative random subset $W$
of the real line.
Complementing our study in previous papers, we identify
self-similar Markovian composition structures associated with the
two-parameter family of partition structures.
@article{ZNSL_2005_326_a5,
author = {A. V. Gnedin and J. Pitman},
title = {Self-similar and {Markov} composition structures},
journal = {Zapiski Nauchnykh Seminarov POMI},
pages = {59--84},
publisher = {mathdoc},
volume = {326},
year = {2005},
language = {en},
url = {http://geodesic.mathdoc.fr/item/ZNSL_2005_326_a5/}
}
A. V. Gnedin; J. Pitman. Self-similar and Markov composition structures. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part XIII, Tome 326 (2005), pp. 59-84. http://geodesic.mathdoc.fr/item/ZNSL_2005_326_a5/