Swap-invariant and exchangeable random measures

In this work we analyze the concept of swap-invariance, which is a weaker variant of exchangeability. A random vector $ξ$ in $\mathbb{R}^n$ is called swap-invariant if $\,{\mathbf E}\,\big| \!\sum_j u_j ξ_j \big|\,$ is invariant under all permutations of $(ξ_1, \ldots, ξ_n)$ for each $u \in \mathbb{R}^n$. We extend this notion to random measures. For a swap-invariant random measure $ξ$ on a measure space $(S,\mathcal{S},μ)$ the vector $(ξ(A_1), \ldots, ξ(A_n))$ is swap-invariant for all disjoint $A_j \in \mathcal{S}$ with equal $μ$-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.

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