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 sp… Show more
“…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.…”
Section: Joint Distributions Of Random Setsmentioning
confidence: 60%
“…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…”
Section: Joint Distributions Of Random Setsmentioning
confidence: 99%
“…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).…”
Section: Joint Distributions Of Random Setsmentioning
confidence: 99%
“…[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).…”
Section: Proposition 1 (Seementioning
confidence: 99%
“…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.…”
“…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.…”
Section: Joint Distributions Of Random Setsmentioning
confidence: 60%
“…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…”
Section: Joint Distributions Of Random Setsmentioning
confidence: 99%
“…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).…”
Section: Joint Distributions Of Random Setsmentioning
confidence: 99%
“…[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).…”
Section: Proposition 1 (Seementioning
confidence: 99%
“…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.…”
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