Nonstandard characterization of convergence in law for $D[0,1]$-valued random variables

Abstract:

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.