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 as earlier, that the limit process is concentrated on the space C[0; 1]; and therefore the theory is now 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