For lower-semicontinuous and convex stochastic processes Zn and nonnegative random variables ϵn we investigate the pertaining random sets A(Zn,ϵn) of all ϵn-approximating minimizers of Zn. It is shown that, if the finite dimensional distributions of the Zn converge to some Z and if the ϵn converge in probability to some constant c, then the A(Zn,ϵn) 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.
Let <span class="tex">ε</span>-Argmin<span class="tex">(Z)</span> be the collection of all <span class="tex">ε</span>-optimal solutions for a stochastic process <span class="tex">Z</span> with locally bounded trajectories defined on a topological space. For sequences <span class="tex">(Z<sub>n</sub>)</span> of such stochastic processes and <span class="tex">(ε<sub>n</sub>)</span> of nonnegative random variables we give sufficient conditions for the (closed) random sets <span class="tex">ε<sub>n</sub></span>-Argmin<span class="tex">(Z<sub>n</sub>)</span> to converge in distribution with respect to the Fell-topology and to the coarser Missing-topology.