An ergodic theorem for partially exchangeable random partitions

We consider shifts Pi(n,m) of a partially exchangeable random partition Pi(infinity) of N obtained by restricting Pi(infinity) to {n + 1; n + 2,..., n + m} and then subtracting n from each element to get a partition of [ m] : = {1,..., 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 N with the same ranked frequencies as Pi(infinity). As a consequence, the partially exchangeable random partition Pi(infinity) is exchangeable if and only if Pi(infinity) 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.