Probability Theory and Related Fields

, Volume 100, Issue 2, pp 175–189

Metric marginal problems for set-valued or non-measurable variables

  • R. M. Dudley
Article
  • 65 Downloads

Summary

In a separable metric space, if two Borel probability measures (laws) are nearby in a suitable metric, then there exist random variables with those laws which are nearby in probability. Specifically, by a well-known theorem of Strassen, the Prohorov distance between two laws is the infimum of Ky Fan distances of random variables with those laws. The present paper considers possible extensions of Strassen's theorem to two random elements one of which may be (compact) set-valued and/or non-measurable. There are positive results in finite-dimensional spaces, but with factors depending on the dimension. Examples show that such factors cannot entirely be avoided, so that the extension of Strassen's theorem to the present situation fails in infinite dimensions.

Mathematics Subject Classifications

60B05 60B10 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [A] Andersen, N.T.: The central limit theorem for non-separable valued functions. Z. Wahrscheinlichkeitstheor. Verw. Geb.70, 445–455 (1985)Google Scholar
  2. [B] Banach, S.: Théorie des opérations linéaires. Warsaw, 1932; Repr. Chelsea, New York (1955)Google Scholar
  3. [D] Dudley, R.M.: Real analysis and probability. 2d printing, corrected. New York London. Chapman and Hall (1993)Google Scholar
  4. [D-P] Dudley, R.M., Philipp, W.: Invariance principles for sums of Banach space valued random elements and empirical processes. Z. Wahrscheinlichkeitstheor. Verw. Geb.62, 509–552 (1983)Google Scholar
  5. [F-L-M] Figiel, T., Lindenstrauss, J., Milman, V.D.: The dimension of almost spherical sections of convex bodies. Acta Math. (Sweden)139 53–94 (1977)Google Scholar
  6. [G-Z] Giné, E., Zinn, J.: Gaussian characterization of uniform Donsker classes of functions. Ann. Probab.19, 758–782 (1991)Google Scholar
  7. [K-R] Kantorovich, L.V., Rubinštein, G.Š.: On a space of completely additive functions. Vestnik Leningrad University1958 no. 7 (Ser. Mat. Mekh. Astron. vyp. 2, 52–59) (in Russian)Google Scholar
  8. [L1] Lévy, Paul: Lecons d'analyse fonctionnelle. Paris: Gauthier-Villars 1922Google Scholar
  9. [L2] L'evy, Paul: Problèmes concrets d'analyse fonctionelle (2d ed. of Lévy, 1922) Paris: Gauthier-Villars 1951Google Scholar
  10. [M] Milman, V.D.: The heritage of P. Lévy in geometrical functional analysis. Astérisque157–158, 273–301 (1988)Google Scholar
  11. [M-S] Milman, V.D., Schechtman, G.: Asymptotic theory of finite dimensional normed spaces, with an Appendix by M. Gromov (Lect. Notes Math., vol. 1200) Berlin Heidelberg New York: Springer 1986Google Scholar
  12. [N] Nachbin, L.: The Haar integral. New York: Van Nostrand 1965Google Scholar
  13. [O] Osserman, R.: The isoperimetric inequality. Bull. Am. Math. Soc.84, 1182–1238 (1978)Google Scholar
  14. [R] Rachev, S.T.: The Monge-Kantorovich mass transference problem and its stochastic applications. Theory Probab. Appl.29, 647–676 (1985)Google Scholar
  15. [R-R-S] Rachev, S.T., Rüschendorf, L., Schief, A.: Uniformities for the convergence in law and in probability. J. Theor. Probab.5, 33–44 (1992)Google Scholar
  16. [S1] Schmidt, E.: Beweis der isoperimetrischen Eigenschaft der Kugel im hyperbolischen und sphärischen Raum jeder Dimensionenzahl. Math. Z.49, 1–109 (1943)Google Scholar
  17. [S2] Schmidt, E.: Die Brunn-Minkowskische Ungleichung und ihr Spiegelbild sowie die isoperimetrische Eigenschaft der Kugel in der euklidischen und nichteuklidischen Geometrie I, II. Math. Nachr.1, 81–157;2, 171–244 (1948, 1949)Google Scholar
  18. [St] Strassen, Volker: The existence of probability measures with given marginals. Ann. Math. Stat.36, 423–439 (1965)Google Scholar
  19. [T] Taylor, A.E.: A geometric theorem and its application to biorthogonal systems. Bull. Am. Math. Soc.53, 614–616 (1947)Google Scholar

Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • R. M. Dudley
    • 1
  1. 1.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA

Personalised recommendations