Non-monotone convergence in the quadratic Wasserstein distance

  • Walter SchachermayerEmail author
  • Uwe Schmock
  • Josef Teichmann
Part of the Lecture Notes in Mathematics book series (LNM, volume 1979)


We give an easy counterexample to Problem 7.20 from C. Villani’s book on mass transport: in general, the quadratic Wasserstein distance between n-fold normalized convolutions of two given measures fails to decrease monotonically.


Random Vector Central Limit Theorem Option Price Austria Email Transport Plan 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    S. Artstein, K. M. Ball, F. Barthe and A. Naor, Solution of Shannon's Problem on the Monotonicity of Entropy, Journal of the AMS 17(4), 2004, pp. 975–982.MathSciNetzbMATHGoogle Scholar
  2. 2.
    L. Bachelier, Théorie de la Spéculation, Annales scientifiques de l'Écöle Normale Supérieure Série 3, 17, 1900, pp. 21–86. Also available from the site MathSciNetzbMATHGoogle Scholar
  3. 3.
    W. Schachermayer, Introduction to the Mathematics of Financial Markets, LNM 1816 - Lectures on Probability Theory and Statistics, Saint-Flour summer school 2000 (Pierre Bernard, editor), Springer-Verlag, Heidelberg, 2003, pp. 111–177.Google Scholar
  4. 4.
    H. Tanaka, An inequality for a functional of probability distributions and its applications to Kac's one-dimensional model of a Maxwell gas, Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 27, 1973, pp. 47–52.zbMATHCrossRefGoogle Scholar
  5. 5.
    C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics 58, American Mathematical Society, Providence Rhode Island, 2003.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Walter Schachermayer
    • 1
    Email author
  • Uwe Schmock
    • 2
  • Josef Teichmann
    • 3
  1. 1.Vienna University of TechnologyWiedner HauptstrasseViennaAustria
  2. 2.Vienna University of TechnologyWiedner HauptstrasseViennaAustria
  3. 3.Vienna University of TechnologyWiedner HauptstrasseViennaAustria

Personalised recommendations