Characterizing joint distributions of random sets by multivariate capacities

Bernhard Schmelzer
2012 International Journal of Approximate Reasoning  
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
more » ... se 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.
doi:10.1016/j.ijar.2012.06.017 fatcat:lr72oty2u5fitmsh5blxtvtkti