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

2025/12/06 by Ferger, Dietmar, Rosar, Niklas
Environmental Science · Mathematics · #Analysis of environmental and stochastic processes #FOS: Mathematics #Fuzzy Systems and Optimization #Point processes and geometric inequalities #Probability (math.PR)

paper · doi:10.48550/arxiv.2512.06375

openalex publication_date 2025/12/06 · openalex created_date 2025/12/10 · openalex updated_date 2026/07/28

Abstract

Consider finitely many nets of multivariate càdlàg stochastic processes. We show that the vectors consisting of the respective minimizing points converge in distribution to a random closed set. This set is given as a cartesian product with factors which are equal to the set of all minimizing points of stochastic processes occurring as functional limits of the respective nets. If these limit processes have almost surely exactly one minimizer, then the vectors converge classically in distribution to the vector of these minimizers.

Citations

• Epi-convergence in distribution of normal integrands with applications to sets of epsilon-optimal solutions
• Weak convergence of probability measures on hyperspaces with the upper Fell-topology
• On semi-continuity and continuity of the smallest and largest minimizing point of real convex functions with applications in probability and statistics
• Asymptotics for minimisers of convex processes
• A continuous mapping theorem for the smallest argmax functional
• Further details on inference under right censoring for transformation models with a change-point based on a covariate threshold

Related

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• Weak convergence of multivariate partial maxima processes
• The argmin process of random walks and Lévy processes
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