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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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


Authors: D. Landers and L. Rogge
Journal: Proc. Amer. Math. Soc. 127 (1999), 199-203
MSC (1991): Primary 28E05; Secondary 60B12
MathSciNet review: 1458252
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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.


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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: http://dx.doi.org/10.1090/S0002-9939-99-04504-9
PII: S 0002-9939(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