Swap-invariant and exchangeable random measures

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 $\mathbf{E}\left|{\sum }_j{u}_j{\xi }_j\right|$ is invariant under all permutations of $({\xi }_1,\dots ,{\xi }_n)$ 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 $(\xi (A_1),\dots ,\xi (A_n))$ 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.

30 pages; v2: variant of ergodic theorem and example added, minor changes in text; v3: structure changed, theorems slightly improved

References in corpus (1)

• Invariance properties of random vectors and stochastic processes based on the zonoid concept