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- W4380993315 abstract "We prove that the smallest maximizer 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 maximizer 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." @default.
- W4380993315 created "2023-06-17" @default.
- W4380993315 creator A5014914729 @default.
- W4380993315 date "2023-06-14" @default.
- W4380993315 modified "2023-09-25" @default.
- W4380993315 title "On semi-continuity and continuity of the smallest and largest maximizing point of real convex functions with applications in probability and statistics" @default.
- W4380993315 doi "https://doi.org/10.48550/arxiv.2306.08358" @default.
- W4380993315 hasPublicationYear "2023" @default.
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