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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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  • [1] Robert M. Anderson, A nonstandard representation for Brownian motion and Itô integration, Israel J. Math. 25 (1976), 15-46.
  • [2] Robert M. Anderson and Salim Rashid, A nonstandard characterization of weak convergence, Proc. Amer. Math. Soc. 69 (1978), 327-332. MR 58:1073
  • [3] Patrick Billingsley, Convergence of probability measures, Wiley, New York and Toronto, 1968. MR 38:1718
  • [4] Dieter Landers and Lothar Rogge, Universal Loeb-measurability of sets and of the standard part map with applications, Trans. Amer. Math. Soc. 304 (1987), 229-243. MR 89d:28015
  • [5] Dieter Landers and Lothar Rogge, Nichtstandard Analysis, Springer Verlag, Heidelberg and New York, 1994. MR 95i:03140
  • [6] Dieter Landers and Lothar Rogge, Nonstandard characterization for a general invariance principle, Advances in analysis, probability and mathematical physics (edited by Albeverio, Luxemburg, Wolff), Kluver Publ. Comp., 1995, pp. 176-185. MR 96i:28024
  • [7] Peter A. Loeb, Conversion from nonstandard to standard measure spaces and applications in probability theory, Trans. Amer. Math. Soc. 211 (1975), 113-122. MR 52:10980
  • [8] Peter A. Loeb, Applications of nonstandard analysis to ideal boundaries in potential theory, Israel J. Math. 25 (1976), 154-187. MR 56:15961
  • [9] Peter A. Loeb, A generalization of the Riesz-Herglotz theorem on representing measures, Proc. Amer. Math. Soc. 71 (1978), 65-68. MR 58:28574
  • [10] Peter A. Loeb, Weak limits of measures and the standard part map, Proc. Amer. Math. Soc. 77 (1979), 128-135. MR 80i:28020
  • [11] Peter A. Loeb, A construction of representing measures for elliptic and parabolic differential equations, Math. Ann. 260 (1982), 51-56. MR 83i:31015
  • [12] G. R. Mendieta, Two hyperfinite constructions of the Brownian bridge, Stochastic Anal. Appl. 7 (1989), 75-88. MR 90e:60047
  • [13] A. Stoll, A nonstandard construction of the Levy Brownian motion, Probab. Theory Related Fields 71 (1986), 321-334. MR 87h:60149
  • [14] K.D. Stroyan and J.M. Bayod, Foundations of infinitesimal stochastic analysis, North-Holland, Amsterdam and New York, 1986. MR 87m:60001

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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

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