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

2025/07/22 by Ferger, Dietmar · 1 citation
#26E25 #60B05 #60B10 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2507.16297

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.

Citations

• Weak convergence of probability measures on hyperspaces with the upper Fell-topology

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• Extended Argmin-Theorems for multiple nets of multivariate càdlàg stochastic processes

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