Epi-convergence almost surely, in probability and in distribution

Petr Lachout ()

Annals of Operations Research, 2006, vol. 142, issue 1, 187-214

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). Copyright Springer Science + Business Media, Inc. 2006

Keywords: Epi-convergence; Convergence almost surely; Convergence in probability; Convergence in distribution; Hit-topology; Miss-topology (search for similar items in EconPapers)
Date: 2006
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DOI: 10.1007/s10479-006-6168-9

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