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A continuous mapping theorem for the argmax-functional in the non-unique case. (English) Zbl 1090.60032
Summary: The Argmax-Continuous Mapping Theorem (Argmax-CMT) of Kim and Pollard resp. van der Vaart and Wellner has been proved to be a very useful tool in statistics for deriving distributional convergence of M-estimators. However it only works as long as the limit process possesses an almost sure unique maximizing point. In this article we prove an extension of the Argmax-CMT where almost sure uniqueness is no longer needed. Moreover our Argmax-CMT is also valid in the function space $$D(\mathbb R)$$ equipped with Lindvall’s version of the Skorokhod topology. As an example the result is applied to change-point estimators.

##### MSC:
 60G07 General theory of stochastic processes 62F10 Point estimation
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