Epi-convergence in distribution of normal integrands with applications to sets of epsilon-optimal solutions

Dietmar Ferger

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arXiv · 2507.16297

Epi-convergence in distribution of normal integrands with applications to sets of epsilon-optimal solutions

Dietmar Ferger

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Abstract

We derive necessary and sufficient conditions for epi-convergence in distribution of normal integrands. As a basic tool for the proof a new characterisation for distributional convergence of random closed sets is used. Our approach via the epi-topology allows us to show that, if a net of normal integrands epiconverges in distribution, then the pertaining sets of epsilon-optimal solutions converge in distribution in the underlying hyperspace endowed with the upper-Fell topology. Under some boundedness and uniquenss assumptions the convergence even holds for the Fell topology. Finally, measurable selections converge weakly to a Choquet-capacity.

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Dietmar Ferger. 2025-07-22. Epi-convergence in distribution of normal integrands with applications to sets of epsilon-optimal solutions. https://arxiv.org/abs/2507.16297

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