Self-similar and Markov composition structures

The bijection between composition structures and random closed subsets of the unit interval implies that the composition structures associated with S ⋂ 0, 1] for a self-similar random set S ⊂ ℝ₊ are those that 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 1’s in the sequence are the positions 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 presented in previous papers, we identify self-similar Markov composition structures associated with the two-parameter family of partition structures. Bibliography: 19 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 326, 2005, pp. 59–84.