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Almost surely convergent random variables with given laws
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  • Published: May 1989

Almost surely convergent random variables with given laws

  • Andreas Schief1 

Probability Theory and Related Fields volume 81, pages 559–567 (1989)Cite this article

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Summary

Let μ n →μ be a weakly converging sequence of Borel probability measures on a topological space X. We prove the existence of an almost surely converging sequence of random variables ξ n →ξ which obey this laws, if a certain μ-dependent countability property of the topology holds. Especially this is the case if

  1. (a)

    X is second countable

  2. (b)

    X is first countable and μ has countable support

  3. (c)

    X is metrizable and μ is τ-smooth.

A final example disproves the existence of such random variables for (tight) measures on a Lusin space.

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References

  1. Beran, R.J., Le Cam, L., Millar, P.W.: Convergence of stochastic empirical measures. J. Multivariate Anal. 23, 159–168 (1987)

    Google Scholar 

  2. Dudley, R.M.: Distances of probability measures and random variables. Ann. Math. Stat. 39, 1563–1572 (1968)

    Google Scholar 

  3. Dudley, R.M.: An extended Wichura theorem, definitions of Donsker class, and weighted empirical distributions. In: Lect. Notes Math. Vol. 1153, pp. 141–178. Berlin Heidelberg New York: Springer 1985

    Google Scholar 

  4. Fernandez, P.J.: Almost surely convergent versions of sequences which converge weakly. Bol. Soc. Bras. Mat. 5, 51–61 (1974)

    Google Scholar 

  5. Fernique, X.: Un modèle presque sûr pour la convergence en loi. C.R. Acad. Sci. Paris, t.306, Série I, 335–338 (1988)

    Google Scholar 

  6. Halmos, P.R.: Measure theory. New York: van Nostrand 1950

    Google Scholar 

  7. Hoffmann-Jørgensen, J.: The general marginal problem. In: Lect. Notes Math., Vol. 1242, pp. 77–367. Berlin Heidelberg New York: Springer 1987

    Google Scholar 

  8. Pyke, R.: Applications of almost surely convergent constructions of weakly convergent processes. In: Lect. Notes Math., Vol. 89, pp. 187–200. Berlin Heidelberg New York: Springer 1969

    Google Scholar 

  9. Schief, A.: On continuous image averaging of Borel measures. To appear in topology and its applications

  10. Schief, A.: Topological properties of the addition map in spaces of Borel measures. Math. Ann. 282, 23–31 (1988)

    Google Scholar 

  11. Schwartz, L.: Radon measures. London: Oxford University Press 1973

    Google Scholar 

  12. Skorokhod, A.V.: Limit theorems for stochastic processes. Theory Probab. Appl. 1, 261–290 (English), 289–319 (Russian) (1956)

    Google Scholar 

  13. Steen, L., Seebach, J.: Counterexamples in topology, 2. ed. New York: Springer 1978

    Google Scholar 

  14. Topsøe F.: Topology and measure (Lect. Notes Math., Vol. 133). Berlin Heidelberg New York: Springer 1970

    Google Scholar 

  15. Wichura, M.: On the construction of almost uniformly convergent random variables with given weakly convergent image laws. Ann. Math. Stat. 41, 284–291 (1970)

    Google Scholar 

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Authors and Affiliations

  1. Mathematisches Institut, Ludwig-Maximilians-Universität, Theresienstr. 39, D-8000, München, Federal Republic of Germany

    Andreas Schief

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  1. Andreas Schief
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Schief, A. Almost surely convergent random variables with given laws. Probab. Th. Rel. Fields 81, 559–567 (1989). https://doi.org/10.1007/BF00367303

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  • Received: 08 August 1988

  • Issue Date: May 1989

  • DOI: https://doi.org/10.1007/BF00367303

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Keywords

  • Stochastic Process
  • Probability Measure
  • Probability Theory
  • Topological Space
  • Statistical Theory
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