Characterizing joint distributions of random sets by multivariate capacities |
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Authors: | Bernhard Schmelzer |
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Institution: | 1. Durham, frank.coolen@durham.ac.uk;2. Innsbruck, thomas.fetz@uibk.ac.at;3. Granada, smc@decsai.ugr.es;4. Innsbruck, michael.oberguggenberger@uibk.ac.at |
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Abstract: | By the Choquet theorem, distributions of random closed sets can be characterized by a certain class of set functions called capacity functionals. In this paper a generalization to the multivariate case is presented, that is, it is proved that the joint distribution of finitely many random sets can be characterized by a multivariate set function being completely alternating in each component, or alternatively, by a capacity functional defined on complements of cylindrical sets. For the special case of finite spaces a multivariate version of the Moebius inversion formula is derived. Furthermore, we use this result to formulate an existence theorem for set-valued stochastic processes. |
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