On semi-continuity and continuity of the smallest and largest minimizing point of real convex functions with applications in probability and statistics

Abstract

We prove that the smallest minimizer s(f) of a real convex function f is less than or equal to a real point x if and only if the right derivative of f at x is non-negative. Similarly, the largest minimizer t(f) is greater or equal to x if and only if the left derivative of f at x is non-positive. From this simple result we deduce measurability and semi-continuity of the functionals s and t. Furthermore, if f has a unique minimizing point, so that s(f) = t(f), then the functional is continuous at f. With these analytical preparations we can apply Continuous Mapping Theorems to obtain several Argmin theorems for convex stochastic processes. The novelty here are statements about classical distributional convergence and almost sure convergence, if the limit process does not have a unique minimum point. This is possible by replacing the natural topology on R with the order topologies. Another new feature is that not only sequences but more generally nets of convex stochastic processes are allowed.