This paper introduces a version of the argmax continuous mapping theorem that applies to M-estimation problems in which the objective functions converge to a limiting process with multiple maximizers. The concept of the smallest maximizer of a function in the d-dimensional Skorohod space is introduced and its main properties are studied. The resulting continuous mapping theorem is applied to three problems arising in change-point regression analysis. Some of the results proved in connection to the d-dimensional Skorohod space are also of independent interest.
Emilio Seijo. Bodhisattva Sen. "A continuous mapping theorem for the smallest argmax functional." Electron. J. Statist. 5 421 - 439, 2011. https://doi.org/10.1214/11-EJS613