Characterizing joint distributions of random sets by multivariate capacities

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.

Joint distributions of random sets and their relation to copulas

“…In [15] it has been shown that similar results hold concerning the joint distribution of finitely many random sets. The aim of this section is to summarize the most important results from [15] without giving the proofs (for proofs the reader is referred to [15]). For the sake of simplicity the considerations are restricted to the two-dimensional case although all statements can be generalized to the n-dimensional case without any problems.…”

“…In [15] the author has shown that in order to characterize the joint distribution of random sets it is favorable to define a multivariate set function that assigns to pairs of compact sets (K 1 , K 2 ) the probability of the event…”

“…Consequently, such a set function shall be called joint (or multivariate) capacity functional. [15].) Let X i : → F i , i = 1, 2, be random closed sets on a probability space ( , , P).…”

“…[15].) Let ψ : K 1 × K 2 → [0, 1] be a joint capacity functional (i.e., a set function satisfying Conditions (i)-(iii) of Proposition 1).…”

“…In Section 2 basic facts about random sets are reviewed. Section 3 summarizes the most important results from [15] concerning joint distributions of random sets and characterization by set functions. Section 4 consists of two parts: Subsection 4.1 is concerned with the question how copulas can be used to describe the dependence of random sets, Subsection 4.2 is addressed to modeling dependence of random sets by using copulas.…”