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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

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

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