This paper establishes the argmin of a random objective function to be unique almost surely. This paper first formulates a general result that proves almost sure uniqueness without convexity of the objective function. The general result is then applied to a variety of applications in statistics. Four applications are discussed, including uniqueness of M-estimators, both classical likelihood and penalized likelihood estimators, and two applications of the argmin theorem, threshold regression and weak identification.
"Almost sure uniqueness of a global minimum without convexity." Ann. Statist. 48 (1) 584 - 606, February 2020. https://doi.org/10.1214/19-AOS1829