Epi-convergence almost surely, in probability and in distribution |
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Authors: | Petr Lachout |
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Institution: | (1) Department of Probability and Statistics, Charles University of Prague, Sokolovská 83, 186 75 Praha 8;(2) Institute of Information Theory and Automation, Czech Academy of Sciences, Pod vodárenskou věží 4, 182 08 Praha 8, Czech Republic |
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Abstract: | The paper deals with an epi-convergence of random real functions defined on a topological space. We follow the idea due to
Vogel (1994) to split the epi-convergence into the lower semicontinuous approximation and the epi-upper approximation and
localize them onto a given set. The approximations are shown to be connected to the miss- resp. hit-part of the ordinary Fell
topology on sets. We introduce two procedures, called “localization”, separately for the miss-topology and the hit-topology
on sets. Localization of the miss- resp. hit-part of the Fell topology on sets allows us to give a suggestion how to define
the approximations in probability and in distribution. It is shown in the paper that in case of the finite-dimensional Euclidean
space, the suggested approximations in probability coincide with the definition from Vogel and Lachout (2003).
The research has been partially supported by Deutsche Forschungsgemeinschaft under grant No. 436TSE113/40, by the Ministry
of Education, Youth and Sports of the Czech Republic under Project MSM 113200008 and by the Grant Agency of the Czech Republic
under grant No. 201/03/1027. |
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Keywords: | Epi-convergence Convergence almost surely Convergence in probability Convergence in distribution Hit-topology Miss-topology |
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