Self-similar and Markov composition structures

TL;DR

This paper explores the connection between self-similar Markov composition structures and stationary regenerative sets, characterizing when Markov chains generate such compositions and linking them to the two-parameter family of partition structures.

Contribution

It characterizes self-similar Markov composition structures via Markov chains and regenerative sets, extending previous work to include the two-parameter family of partition structures.

Findings

Markov chains correspond to self-similar composition structures when associated with exponential of stationary regenerative sets.

Identifies conditions under which composition structures are consistent with simple truncation.

Extends the theory to include the two-parameter family of partition structures.

Abstract

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.