Geometry of the Expected Value Set and the Set-Valued Sample Mean Process
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The law of large numbers extends to random sets by employing Minkowski addition. Above that, a central limit theorem is available for set-valued random variables. The existing results use abstract isometries to describe convergence of the sample mean process toward the limit, the expected value set. These statements do not reveal the local geometry and the relations of the sample mean and the expected value set, so these descriptions are not entirely satisfactory in understanding the limiting behavior of the sample mean process. This paper addresses and describes the fluctuations of the sample average mean on the boundary of the expectation set.
KeywordsRandom sets Set-valued integration Stochastic optimization Set-valued risk measures
Mathematics Subject Classification (2010)90C15 26E25 49J53 28B20
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Special thanks to Prof. Roger J.-B. Wets and Prof. Georg Ch. Pflug, who encouraged and supported investigating set-valued mappings. Both provided useful comments on initial versions of this paper.
We would like to thank the editor of the journal and two independent referees for their commitment to assess the paper. Their comments were very professional and profound and lead to a significant improvement of the content.
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