Nonstandard methods for families of -smooth probability measures

Authors:
Dieter Landers and Lothar Rogge

Journal:
Proc. Amer. Math. Soc. **103** (1988), 1151-1156

MSC:
Primary 28E05; Secondary 28A33, 60B10

DOI:
https://doi.org/10.1090/S0002-9939-1988-0954998-8

MathSciNet review:
954998

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Abstract | References | Similar Articles | Additional Information

Abstract: For families of -smooth probability measures we give nonstandard characterizations of uniform -smoothness, uniform tightness, uniform pretightness, and relative compactness in the weak topology. We apply these characterizations to obtain two important theorems of probability theory: A theorem of Topsøe and a theorem of Prohorov.

**[1]**Robert M. Anderson and Salim Rashid,*A nonstandard characterization of weak convergence*, Proc. Amer. Math. Soc.**69**(1978), no. 2, 327–332. MR**0480925**, https://doi.org/10.1090/S0002-9939-1978-0480925-X**[2]**Patrick Billingsley,*Convergence of probability measures*, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR**0233396****[3]**Nigel J. Cutland,*Nonstandard measure theory and its applications*, Bull. London Math. Soc.**15**(1983), no. 6, 529–589. MR**720746**, https://doi.org/10.1112/blms/15.6.529**[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.**[5]**Peter A. Loeb,*Conversion from nonstandard to standard measure spaces and applications in probability theory*, Trans. Amer. Math. Soc.**211**(1975), 113–122. MR**0390154**, https://doi.org/10.1090/S0002-9947-1975-0390154-8**[6]**D. W. Müller,*Nonstandard proofs of invariance principles in probability theory*, Applications of Model Theory to Algebra, Analysis, and Probability (Internat. Sympos., Pasadena, Calif., 1967) Holt, Rinehart and Winston, New York, 1969, pp. 186–194. MR**0239645****[7]**K. D. Stroyan and W. A. J. Luxemburg,*Introduction to the theory of infinitesimals*, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. Pure and Applied Mathematics, No. 72. MR**0491163****[8]**Flemming Topsøe,*Compactness and tightness in a space of measures with the topology of weak convergence*, Math. Scand.**34**(1974), 187–210. MR**0388484**, https://doi.org/10.7146/math.scand.a-11520

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1988-0954998-8

Keywords:
Uniformly -smooth,
uniformly tight,
weakly relatively compact families of -measures,
Loeb-measures,
theorem of Prohorov and Topsøe

Article copyright:
© Copyright 1988
American Mathematical Society