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The distribution of unbounded random sets and the multivalued strong law of large numbers in nonreflexive Banach spaces. (English) Zbl 0945.60024
Summary: In the first part, we introduce appropriate tools concerning the distribution of random sets. We study the relation between the distribution of a random set, whose values are closed subsets of a Banach space, and the set of distributions of its measurable selections. Also, criteria for two random sets to be equidistributed are given, along with applications to the multivalued integral. In the second part, in combination with other arguments involving convex analysis and topological properties of hyperspaces (i.e., spaces of subsets), the results of the first part are exploited to prove a multivalued strong law of large numbers for closed (possibly unbounded) valued random sets, when the space of all closed sets is endowed, either with the Wijsman topology or the ‘slice topology’ introduced by G. Beer. The main results extend others of the same type in the literature, especially in the framework of non-reflexive Banach space, or allow for shorter and self-contained proofs.

MSC:
60F15 Strong limit theorems
60E99 Distribution theory
60D05 Geometric probability and stochastic geometry
26E25 Set-valued functions
28B20 Set-valued set functions and measures; integration of set-valued functions; measurable selections
52A05 Convex sets without dimension restrictions (aspects of convex geometry)
52A22 Random convex sets and integral geometry (aspects of convex geometry)
54C60 Set-valued maps in general topology
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