Geodesic
A continuous mapping theorem for the argmin-set functional with applications to convex stochastic processes
Kybernetika, Tome 57 (2021) no. 3, pp. 426-445 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
DOI : 10.14736/kyb-2021-3-0426
Classification : 60B05, 60B10, 60F99
Keywords: convex stochastic processes; sets of approximating minimizers; weak convergence; Vietoris hyperspace topologies; Choquet-capacity 
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     author = {Ferger, Dietmar},
     title = {A continuous mapping theorem for the argmin-set functional with applications to convex stochastic processes},
     journal = {Kybernetika},
     pages = {426--445},
     year = {2021},
     volume = {57},
     number = {3},
     doi = {10.14736/kyb-2021-3-0426},
     mrnumber = {4299457},
     zbl = {07442518},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.14736/kyb-2021-3-0426/}
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Ferger, Dietmar. A continuous mapping theorem for the argmin-set functional with applications to convex stochastic processes. Kybernetika, Tome 57 (2021) no. 3, pp. 426-445. doi: 10.14736/kyb-2021-3-0426

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