Swap-invariant and exchangeable random measures

TL;DR

This paper introduces and analyzes swap-invariance, a weaker form of exchangeability for random measures, providing characterizations, ergodic theorems, and new representations, with implications for understanding the structure of such measures.

Contribution

It extends the concept of swap-invariance to random measures, establishes their properties, and connects them to exchangeable measures through ergodic theorems and representations.

Findings

Swap-invariant measures are characterized and related to exchangeable measures.

Ergodic theorems are proved for swap-invariant measures.

Diffuse swap-invariant measures on Borel spaces are trivial.

Abstract

In this work we analyze the concept of swap-invariance, which is a weaker variant of exchangeability. A random vector $\xi$ in $\mathbb{R}^n$ is called swap-invariant if $\,{\mathbf E}\,\big| \!\sum_j u_j \xi_j \big|\,$ is invariant under all permutations of $(\xi_1, \ldots, \xi_n)$ for each $u \in \mathbb{R}^n$. We extend this notion to random measures. For a swap-invariant random measure $\xi$ on a measure space $(S,\mathcal{S},\mu)$ the vector $(\xi(A_1), \ldots, \xi(A_n))$ is swap-invariant for all disjoint $A_j \in \mathcal{S}$ with equal $\mu$-measure. Various characterizations of swap-invariant random measures and connections to exchangeable ones are established. We prove the ergodic theorem for swap-invariant random measures and derive a representation in terms of the ergodic limit and an exchangeable random measure. Moreover we show that diffuse swap-invariant random measures on a Borel space are trivial. As for random sequences two new representations are obtained using different ergodic limits.