  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.
