Epi-convergence in distribution of normal integrands with applications to sets of epsilon-optimal solutions
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866918101563473920 |
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| author | Ferger, Dietmar |
| author_facet | Ferger, Dietmar |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_16297 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Epi-convergence in distribution of normal integrands with applications to sets of epsilon-optimal solutions Ferger, Dietmar Probability 60B05, 60B10, 26E25 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. |
| title | Epi-convergence in distribution of normal integrands with applications to sets of epsilon-optimal solutions |
| topic | Probability 60B05, 60B10, 26E25 |
| url | https://arxiv.org/abs/2507.16297 |