Swap-invariant and exchangeable random sequences and measures
نویسنده
چکیده
In this work we analyze the concept of swap-invariance, which is a weaker variant of exchangeability. An integrable random vector ξ in R is called swap-invariant if E ∣∣∑ j ujξj ∣∣ is invariant under all permutations of the components of ξ for each u ∈ R. 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 swapinvariance to random measures. A swap-invariant random measure ξ on a measure space (S,S, μ) has the property that (ξ(A1), . . . , ξ(An)) is swapinvariant for all disjoint Aj ∈ S with equal μ-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 swapinvariant random measures on a Borel space.
منابع مشابه
Exchangeable Variable Models
A sequence of random variables is exchangeable if its joint distribution is invariant under variable permutations. We introduce exchangeable variable models (EVMs) as a novel class of probabilistic models whose basic building blocks are partially exchangeable sequences, a generalization of exchangeable sequences. We prove that a family of tractable EVMs is optimal under zeroone loss for a large...
متن کاملComputable Exchangeable Sequences Have Computable de Finetti Measures
We prove a uniformly computable version of de Finetti’s theorem on exchangeable sequences of real random variables. In the process, we develop machinery for computably recovering a distribution from its sequence of moments, which suffices to prove the theorem in the case of (almost surely) continuous directing random measures. In the general case, we give a proof inspired by a randomized algori...
متن کاملOn the geometry of a class of invariant measures and a problem of Aldous
In his survey [4] of notions of exchangeability, Aldous introduced a form of exchangeability corresponding to the symmetries of the infinite discrete cube, and asked whether these exchangeable probability measures enjoy a representation theorem similar to those for exchangeable sequences [11], arrays [12, 13, 1, 2] and set-indexed families [15]. In this note we to prove that, whereas the known ...
متن کاملThe Class of Random Graphs Arising from Exchangeable Random Measures
We introduce a class of random graphs that we argue meets many of the desiderata one would demand of a model to serve as the foundation for a statistical analysis of real-world networks. The class of random graphs is defined by a probabilistic symmetry: invariance of the distribution of each graph to an arbitrary relabelings of its vertices. In particular, following Caron and Fox, we interpret ...
متن کاملRepresentation theorems for partially exchangeable random variables
We provide representation theorems for both finite and countable sequences of finite-valued random variables that are considered to be partially exchangeable. In their most general form, our results are presented in terms of sets of desirable gambles, a very general framework for modelling uncertainty. Its key advantages are that it allows for imprecision, is more expressive than almost every o...
متن کامل