The argmax theorem is a useful result for deriving the limiting distribution
of estimators in many applications. The conclusion of the argmax theorem states
that the argmax of a sequence of stochastic processes converges in distribution
to the argmax of a limiting stochastic process. This paper generalizes the
argmax theorem to allow the maximization to take place over a sequence of
subsets of the domain. If the sequence of subsets converges to a limiting
subset, then the conclusion of the argmax theorem continues to hold. We
demonstrate the usefulness of this generalization in three applications:
estimating a structural break, estimating a parameter on the boundary of the
parameter space, and estimating a weakly identified parameter. The generalized
argmax theorem simplifies the proofs for existing results and can be used to
prove new results in these literatures