An ergodic theorem for partially exchangeable random partitions

We consider shifts $\Pi_{n,m}$ of a partially exchangeable random partition $\Pi_\infty$ of $\mathbb{N}$ obtained by restricting $\Pi_\infty$ to $\{n+1,n+2,\dots, n+m\}$ and then subtracting $n$ from each element to get a partition of $[m]:= \{1, \ldots, m \}$. We show that for each fixed $m$ the distribution of $\Pi_{n,m}$ converges to the distribution of the restriction to $[m]$ of the exchangeable random partition of $\mathbb{N}$ with the same ranked frequencies as $\Pi_\infty$. As a consequence, the partially exchangeable random partition $\Pi_\infty$ is exchangeable if and only if $\Pi_\infty$ is stationary in the sense that for each fixed $m$ the distribution of $\Pi_{n,m}$ on partitions of $[m]$ is the same for all $n$. We also describe the evolution of the frequencies of a partially exchangeable random partition under the shift transformation. For an exchangeable random partition with proper frequencies, the time reversal of this evolution is the heaps process studied by Donnelly and others.