Extended Argmin-Theorems for multiple nets of multivariate c\`adl\`ag stochastic processes

TL;DR

This paper extends argmin theorems to multiple nets of multivariate cdlg stochastic processes, showing convergence of minimizers to a random closed set and classical convergence when limits are unique.

Contribution

It introduces extended argmin theorems for multiple nets of multivariate cdlg processes, characterizing convergence of minimizers in distribution.

Findings

Vectors of minimizers converge to a random closed set.

If limit processes have unique minimizers, convergence is classical.

Provides a framework for analyzing convergence of stochastic process minimizers.

Abstract

Consider finitely many nets of multivariate c\`adl\`ag 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.