A consistent Markov partition process generated from the paintbox process

Abstract

We study a family of Markov processes on ${\mathcal{P}}^{(k)}$, the space of partitions of the natural numbers with at most $k$ blocks. The process can be constructed from a Poisson point process on ${\mathbb{R}}^{+}\times {\prod }_{i=1}^k{\mathcal{P}}^{(k)}$ with intensity $dt\otimes {\varrho }_{\nu }^{(k)}$, where ${\varrho }_{\nu }$ is the distribution of the paintbox based on the probability measure $\nu$ on $\masspartition$, the set of ranked-mass partitions of 1, and ${\varrho }_{\nu }^{(k)}$ is the product measure on ${\prod }_{i=1}^k{\mathcal{P}}^{(k)}$. We show that these processes possess a unique stationary measure, and we discuss a particular set of reversible processes for which transition probabilities can be written down explicitly.

20 pages; J. Appl. Probab. 2011, 48 (3)