 Global argmin may escape a compact convergence set

Uniform convergence in probability of empirical criteria on a compact set proves consistency for minimizers in that set; it does not confine a minimizer over a larger domain. A bounded identifiable regression function can approach its true value again at infinity. For median regression, $a(\theta )={\theta }^2/(1+{\theta }^4)$ has unique zero at zero, but a small positive fitted sample median $c$ has a second solution ${\theta }^2=(1+\sqrt{1-4c^2})/(2c)$. This solution diverges as $c\downarrow 0$, although uniform convergence on any fixed compact interval holds. Constraining the estimator, or proving suitable global separation and localization, repairs the argument.