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

Ferger, Dietmar

Kybernetika, Tome 57 (2021) no. 3, pp. 426-445

Abstract (VO)

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.

Détail

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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