Swap-invariant and exchangeable random sequences and measures

In this work we analyze the concept of swap-invariance, which is a weaker variant of exchangeability. An integrable 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 the components of $\xi$ for each $u \in \mathbb{R}^n$. Further a random sequence is swap-invariant if its finite-dimensional distributions are swap-invariant. Two characterizations of large classes of swap-invariant sequences are given in terms of their ergodic limits and exchangeable sequences. We extend the theory of swap-invariance to random measures. A swap-invariant random measure $\xi$ on a measure space $(S,\mathcal{S},\mu)$ has the property that $(\xi(A_1), \ldots, \xi(A_n))$ is swap-invariant for all disjoint $A_j \in \mathcal{S}$ with equal $\mu$-measure. Various characterizations and connections to exchangeable random measures are established. As major results we obtain an ergodic theorem for swap-invariant random measures on general measure spaces and a characterization of diffuse swap-invariant random measures on a Borel space.