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Title: A continuous mapping theorem for the argmin-set functional with applications to convex stochastic processes (English)
Author: Ferger, Dietmar
Language: English
Journal: Kybernetika
ISSN: 0023-5954 (print)
ISSN: 1805-949X (online)
Volume: 57
Issue: 3
Year: 2021
Pages: 426-445
Summary lang: English
Category: math
Summary: For lower-semicontinuous and convex stochastic processes $Z_n$ and nonnegative random variables $\epsilon_n$ we investigate the pertaining random sets $A(Z_n,\epsilon_n)$ of all $\epsilon_n$-approximating minimizers of $Z_n$. It is shown that, if the finite dimensional distributions of the $Z_n$ converge to some $Z$ and if the $\epsilon_n$ converge in probability to some constant $c$, then the $A(Z_n,\epsilon_n)$ converge in distribution to $A(Z,c)$ in the hyperspace of Vietoris. As a simple corollary we obtain an extension of several argmin-theorems in the literature. In particular, in contrast to these argmin-theorems we do not require that the limit process has a unique minimizing point. In the non-unique case the limit-distribution is replaced by a Choquet-capacity. (English)
Keyword: convex stochastic processes
Keyword: sets of approximating minimizers
Keyword: weak convergence
Keyword: Vietoris hyperspace topologies
Keyword: Choquet-capacity
MSC: 60B05
MSC: 60B10
MSC: 60F99
idZBL: Zbl 07442518
idMR: MR4299457
DOI: 10.14736/kyb-2021-3-0426
Date available: 2021-11-04T12:43:53Z
Last updated: 2022-02-24
Stable URL:
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