A continuous mapping theorem for the argmin-set functional with applications to convex stochastic processes

Dietmar Ferger

Kybernetika (2021)

  • Volume: 57, Issue: 3, page 426-445
  • ISSN: 0023-5954

Abstract

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For lower-semicontinuous and convex stochastic processes and nonnegative random variables we investigate the pertaining random sets of all -approximating minimizers of . It is shown that, if the finite dimensional distributions of the converge to some and if the converge in probability to some constant , then the converge in distribution to 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.

How to cite

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Ferger, Dietmar. "A continuous mapping theorem for the argmin-set functional with applications to convex stochastic processes." Kybernetika 57.3 (2021): 426-445. <http://eudml.org/doc/297471>.

@article{Ferger2021,
abstract = {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.},
author = {Ferger, Dietmar},
journal = {Kybernetika},
keywords = {convex stochastic processes; sets of approximating minimizers; weak convergence; Vietoris hyperspace topologies; Choquet-capacity},
language = {eng},
number = {3},
pages = {426-445},
publisher = {Institute of Information Theory and Automation AS CR},
title = {A continuous mapping theorem for the argmin-set functional with applications to convex stochastic processes},
url = {http://eudml.org/doc/297471},
volume = {57},
year = {2021},
}

TY - JOUR
AU - Ferger, Dietmar
TI - A continuous mapping theorem for the argmin-set functional with applications to convex stochastic processes
JO - Kybernetika
PY - 2021
PB - Institute of Information Theory and Automation AS CR
VL - 57
IS - 3
SP - 426
EP - 445
AB - 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.
LA - eng
KW - convex stochastic processes; sets of approximating minimizers; weak convergence; Vietoris hyperspace topologies; Choquet-capacity
UR - http://eudml.org/doc/297471
ER -

References

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