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.
$\epsilon -$argmin of stochastic process, random closed sets, weak convergence of Hoffmann-Jørgensen, Fell-topology, Missing-topology
49J53, 60B10, 60F05, 90C15