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 ∈_n we investigate the pertaining random sets A(Z_n, ∈_n) of all ∈_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 ∈_n converge in probability to some constant c, then the A(Z_n, ∈_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. [ABSTRACT FROM AUTHOR]