The distribution of unbounded random sets and the multivalued strong law of large numbers in nonreflexive Banach spaces.

*(English)*Zbl 0945.60024Summary: 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 |