Self-similar and Markov composition 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.

16 pages

References in corpus (1)

• Regenerative Compositions in the Case of Slow Variation