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Keywords:
$\epsilon$-argmin of stochastic process; random closed sets; weak convergence of Hoffmann--Jørgensen; Fell-topology; Missing-topology
$\epsilon$-argmin of stochastic process; random closed sets; weak convergence of Hoffmann--Jørgensen; Fell-topology; Missing-topology
Summary:
Let $\epsilon-\text(Z)$ be the collection of all $\epsilon$-optimal solutions for a stochastic process $Z$ with locally bounded trajectories defined on a topological space. For sequences $(Z_n)$ of such stochastic processes and $(\epsilon_n)$ of nonnegative random variables we give sufficient conditions for the (closed) random sets $\epsilon_n-\text(Z_n)$ to converge in distribution with respect to the Fell-topology and to the coarser Missing-topology.
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