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Keywords:
convex stochastic processes; sets of approximating minimizers; weak convergence; Vietoris hyperspace topologies; Choquet-capacity
convex stochastic processes; sets of approximating minimizers; weak convergence; Vietoris hyperspace topologies; Choquet-capacity
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.
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