A strong law of large numbers for non identically distributed Pettis integrable random sets in a general Banach space. (English) Zbl 1519.60005
The aim of this paper is to prove a strong law of large numbers (SLLN) for a sequence of pairwise independent Pettis integrable random sets with values in the family of convex closed subsets of a Banach space \(E\) without any geometric condition on \(E\). The authors provide a multivalued version of the Cesàro mean convergence theorem and some preliminary lemmas which allow to prove some SLLN for pairwise independent Pettis integrable random sets getting various convergence results such as Mosco convergence, weak convergence and linear convergence. Then the authors study the Mosco convergence and Wijsman convergence of the centered sequence created from independent Pettis integrable random sets.
Reviewer: Yuliya S. Mishura (Kyïv)
MSC:
| 60B12 | Limit theorems for vector-valued random variables (infinite-dimensional case) |
| 60E05 | Probability distributions: general theory |
| 60F15 | Strong limit theorems |
| 60G50 | Sums of independent random variables; random walks |
Keywords:
Pettis integration; strong law of large numbers; random sets; Cesàro mean convergence theoremReferences:
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