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

For lower-semicontinuous and convex stochastic processes ${Z}_{n}$ and nonnegative random variables ${\u03f5}_{n}$ we investigate the pertaining random sets $A({Z}_{n},{\u03f5}_{n})$ of all ${\u03f5}_{n}$-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 ${\u03f5}_{n}$ converge in probability to some constant $c$, then the $A({Z}_{n},{\u03f5}_{n})$ 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...