Journal of Convex Analysis 06 (1999), No. 1, 163--182
Copyright Heldermann Verlag 1999
The Distribution of Unbounded Random Sets and the Multivalued Strong Law of Large Numbers in Nonreflexive Banach Spaces
Christian Hess
Centre Viabilité, Jeux, Contrôle, Université Paris IX Dauphine, Place de Lattre de Tassigny, 75775 Paris, France
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.
Keywords: Distribution of random sets, multivalued strong law of large numbers, set convergence, measurable multifunctions, convex sets.
MSC: 26E25; 28B20, 52A05, 52A22, 54C60, 60F15, 60E99
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