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A continuous mapping theorem for the argmin-set functional with applications to convex stochastic processes. (English) Zbl 07442518

Summary: For lower-semicontinuous and convex stochastic processes \(Z_n\) and nonnegative random variables \(\epsilon_n\) we investigate the pertaining random sets \(A(Z_n,\epsilon_n)\) of all \(\epsilon_n\)-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 \(\epsilon_n\) converge in probability to some constant \(c\), then the \(A(Z_n,\epsilon_n)\) 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.

MSC:

60B05 Probability measures on topological spaces
60B10 Convergence of probability measures
60F99 Limit theorems in probability theory
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