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A continuity theorem for cores of random closed sets


Author: Pedro Terán
Journal: Proc. Amer. Math. Soc. 136 (2008), 4417-4425
MSC (2000): Primary 60D05
DOI: https://doi.org/10.1090/S0002-9939-08-09412-4
Published electronically: July 18, 2008
MathSciNet review: 2431058
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Abstract: If a sequence of random closed sets $ X_n$ in a separable complete metric space converges in distribution in the Wijsman topology to $ X$, then the corresponding sequence of cores (sets of probability measures dominated by the capacity functional of $ X_n$) converges to the core of the capacity of $ X$. Core convergence is achieved not only in the Wijsman topology, but even in the stronger Vietoris topology. This is a generalization for unbounded random sets of the result proved by Artstein for random compact sets using the Hausdorff metric.


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Additional Information

Pedro Terán
Affiliation: Facultad de Ciencias Económicas y Empresariales, Grupo Decisión Multicriterio Zaragoza, Universidad de Zaragoza, Gran Vía 2, E-50005 Zaragoza, Spain
Address at time of publication: Miguel Servet 2, $2^{∘}$D, E-33213, Gijón, Spain
Email: teran@unizar.es

DOI: https://doi.org/10.1090/S0002-9939-08-09412-4
Keywords: Choquet capacity, core, random closed set, Vietoris topology, Wijsman topology
Received by editor(s): December 29, 2006
Received by editor(s) in revised form: October 25, 2007
Published electronically: July 18, 2008
Additional Notes: This research was partially supported by Spain’s Ministerio de Educación y Ciencia under its research grants MTM2005-02254 and TSI2005-02511, and the Gobierno de Aragón under its research grant PM2004-052.
Communicated by: Richard C. Bradley
Article copyright: © Copyright 2008 Pedro Terán

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