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Random capacities and their distributions. (English) Zbl 0581.60042
We formalize the notion of an increasing and outer continuous random process, indexed by a class of compact sets, that map the empty set on zero. Existence and convergence theorems for distributions of such processes are proved. These results generalize or are similar to those known in the special cases of random measures, random (closed) sets and random (upper) semicontinuous functions. For the latter processes infinite divisibility under the maximum is introduced and characterized.
Our result generalizes known characterizations of infinite divisibility for random sets and max-infinite divisibility for random vectors. Also discussed is the convergence in distribution of the row-wise maxima of a null-array of random semicontinuous functions.

MSC:
60G99 Stochastic processes
60E07 Infinitely divisible distributions; stable distributions
60F05 Central limit and other weak theorems
60G57 Random measures
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