Nonstandard Characterization of Convergence in Law for D[0,1]-Valued Random Variables
We prove for random variables with values in the space D[0,1] of cadlag functions - endowed with the supremum metric - that convergence in law is equivalent to nonstandard constructions of internal S-cadlag processes, which represent up to an infinitesimal error the limit process. It is not required that the limit process is concentrated on the space C[0,1], so that the theory is applicable to a wider class of limit processes as e.g. to Poisson processes or Gaussian processes. If we consider in D[0,1] the Skorokhod metric - instead of the supremum metric - we obtain a corresponding equivalence to constructions of internal processes with S-separated jumps. We apply these results to functional central limit theorems.
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