Almost Sure Uniqueness of a Global Minimum Without Convexity
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This paper provides a theorem for the set of global minimizers, the argmin, of a random objective function to be unique almost surely. The usual way to get uniqueness is to assume the function is strictly quasiconvex and the domain is convex. Outside of a few special cases, verifying uniqueness without convexity has not been done and is often just assumed. The main result of this paper establishes uniqueness without assuming convexity by relying on an easy-to-verify nondegeneracy condition. The main result of this paper has widespread application beyond econometrics. Six applications are discussed: uniqueness of M-estimators, utility maximization with a nonconvex budget set, uniqueness of the policy function in dynamic programming, envelope theorems, limit theory in weakly identified models, and functionals of Brownian motion.
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