Characterizing joint distributions of random sets by multivariate capacities
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 he 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. (C) 2012 Elsevier Inc. All rights reserved.
joint distribution random set choquet theorem capacity functional moebius inversion formula daniell-kolmogorov theorem