Nonstandard characterization

of convergence in law

for -valued random variables

Authors:
D. Landers and L. Rogge

Journal:
Proc. Amer. Math. Soc. **127** (1999), 199-203

MSC (1991):
Primary 28E05; Secondary 60B12

DOI:
https://doi.org/10.1090/S0002-9939-99-04504-9

MathSciNet review:
1458252

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Abstract | References | Similar Articles | Additional Information

Abstract: We prove for random variables with values in the space of cadlag functions - endowed with the supremum metric - that convergence in law is equivalent to nonstandard constructions of internal -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 , 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 the Skorokhod metric - instead of the supremum metric - we obtain a corresponding equivalence to constructions of internal processes with -separated jumps. We apply these results to functional central limit theorems.

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Additional Information

**D. Landers**

Affiliation:
Dieter Landers, Mathematisches Institut der Universität zu Köln, Weyertal 86, D–50931 Köln, Germany

Email:
landers@mi.uni-koeln.de

**L. Rogge**

Affiliation:
Lothar Rogge, Fachbereich Mathematik der Gerhard-Mercator-Universität ghs Duisburg, Lotharstr. 65, D–47048 Duisburg, Germany

Email:
rogge@math.uni-duisburg.de

DOI:
https://doi.org/10.1090/S0002-9939-99-04504-9

Keywords:
Convergence in law for processes,
nonstandard characterization

Received by editor(s):
February 3, 1997

Received by editor(s) in revised form:
May 8, 1997

Communicated by:
Stanley Sawyer

Article copyright:
© Copyright 1999
American Mathematical Society