Commentary on Probabilistic Geometry

  • B. Schweizer


In our thirty-plus year association with Karl Menger, neither Abe Sklar nor I ever thought of asking him how, in 1942, he came to the idea of a “statistical metric”, i.e., of replacing the classical numerical-valued distance d(p, q) between two points p and q by a probability distribution function Fpq whose value Fpq(x) is to be interpreted as the probability that the “distance” between p and q is less than x. It could have been stimulated by his Rice Institute lecture “Topology without Points” [40] or, more likely, by the lectures on “The Principles of Statistical Inference” which his close friend and former student, Abraham Wald, gave at the University of Notre Dame in 1941 [78]. Or it could have just come to him; after all, he was a man who was constantly getting new ideas.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    J. Aczél: Lectures on Functional Equations and Their Applications, Academic Press, New York, 1966.zbMATHGoogle Scholar
  2. 2.
    C. Alsina, M. J. Frank, B. Schweizer: Associative Functions on Intervals, to appear. Commentary on Probabilistic GeometryGoogle Scholar
  3. 3.
    C. Alsina, B. Schweizer, A. Sklar: On the definition of a probabilistic normed space. Aeq. Math. 46 (1993) 91–96.MathSciNetzbMATHGoogle Scholar
  4. 4.
    C. Alsina, B. Schweizer, C. Sempi, A. Sklar: On the definition of a probabilistic inner product space. Rend. Mat. (7) (1997) 17, 115–127.MathSciNetzbMATHGoogle Scholar
  5. 5.
    A. Appert, Ky Fan: Espaces topologiques intermédiaires. Actualitiés Sci. Ind. 1121, Hermann et Cie, Paris, 1951.zbMATHGoogle Scholar
  6. 6.
    F. Balibrea, B. Schweizer, A. Sklar, J. Smítal: On the generalized specification property and distributional chaos. Intern. J. of Bifurcation and Chaos, to appear.Google Scholar
  7. 7.
    R. Bellman, W. Karush: On the maximum transform. J. Math. Anal. Appl. 6 (1963) 67–74.CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    V. Benes, J. Stépán (eds.): Distributions with Given Marginals and Moment Problems, Kluwer, Dordrecht, 1997.zbMATHGoogle Scholar
  9. 9.
    D. I. Blokhintsev: Space and Time in the Microworld, D. Reidel, Dordrecht, 1973.Google Scholar
  10. 10.
    H. F. Bohnenblust: An axiomatic characterization of La-spaces. Duke Math. J. 6 (1940) 627–640.CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    L. deBroglie: Une remarque sur l’interaction entre la matiére et le champ electro-magnétique. C. R. Acad. Sci. Paris 200 (1935) 361–363.zbMATHGoogle Scholar
  12. 12.
    P. Capéraà, A.-L. Fougéres, C. Genest: Bivariate distributions with given extreme value attractor. J. Multivariate Anal. 72 (2000) 30–49.CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    E. Čech: Topological Spaces, Wiley, New York, 1966.zbMATHGoogle Scholar
  14. 14.
    C. M. Cuadras, J. Fortiana, J. A. Rodriguez-Lallena (eds.): Distributions with Given Marginals and Statistical Modelling, Kluwer, Dordrecht, to appear.Google Scholar
  15. 15.
    G. Dall’Aglio, S. Kotz, G. Salinetti (eds.): Advances in Probability Distribution Functions with Given Marginals: Beyond the Copulas, Mathematics and its Applications, v. 67. Kluwer, Dordrecht, 1991.Google Scholar
  16. 16.
    W. F. Darsow, B. Nguyen, E. T. Olsen: Copulas and Markov processes. Ill. J. Math. 36 (1992) 600–642.MathSciNetzbMATHGoogle Scholar
  17. 17.
    P. Deheuvels: A nonparametric test for independence. Pub. Inst. Statist. Univ. Paris 26 (1981) 29–50.zbMATHGoogle Scholar
  18. 18.
    P. Deheuvels: Probabilistic aspects of multivariate extremes, in Statistical Extremes and Applications, ed. by J. Tiago de Oliveira, D. Reidel, Dordrecht, 1984, 117–130.Google Scholar
  19. 19.
    E. F. Diday, R. Emilion, Y. Hillali: Symbolic data analysis of probabilistic objects by capacities and credibilities, Atti della XXXVIII Riunione Societa Italiana di Statistica, Rimini, 1996, 5–22.Google Scholar
  20. 20.
    E. F. Diday: Symbolic data analysis and the SODAS project; purpose, history, perspective, in Analysis of Symbolic Data, ed. by H.-H. Bock, E. F. Diday, Springer, New York, 2000, 1–23.zbMATHGoogle Scholar
  21. 21.
    A. S. Eddington: Fundamental Theory, Cambridge Univ. Press, London, 1953.Google Scholar
  22. 22.
    T. Erber, B. Schweizer, A. Sklar: Mixing transformations on metric spaces. Comm. Math. Phys. 29 (1973) 311–317.CrossRefMathSciNetzbMATHGoogle Scholar
  23. 23.
    G. L. Forti, L. Paganoni, J. Smital: Dynamics of homeomorphisms on minimal sets generated by triangular mappings. Bull. Austral. Math. Soc. 59 (1999) 1–20.CrossRefMathSciNetzbMATHGoogle Scholar
  24. 24.
    M. J. Frank: Associativity in a class of operations on a space of distribution functions. Aeq. Math. 12 (1975) 121–144.CrossRefMathSciNetzbMATHGoogle Scholar
  25. 25.
    M. Fréchet: Les éléments aléatoires de nature quelconque dans un espace distancié. Ann. Inst. H. Poincare 10 (1948) 215–310.MathSciNetzbMATHGoogle Scholar
  26. 26.
    C. Genest, L.-P. Rivest: Statistical inference procedures for bivariate Archimedean copulas. J. Amer. Statist. Assoc. 88 (1993) 1034–1043.CrossRefMathSciNetzbMATHGoogle Scholar
  27. 27.
    Guo Tie-xin: Survey of recent developments of random metric theory and its applications in China. Acta Analysis Functionalis Applicata 3 (2001) 129–158 and 208–229.Google Scholar
  28. 28.
    U. Höhle: The Poincaré Paradox and non-classical logics, in Fuzzy Sets, Logics and Reasoning about Knowledge, ed. by D. Dubois et al., Kluwer, Dordrecht, 1999, 7–16.Google Scholar
  29. 29.
    U. Höhle, S. E. Rodabaugh (eds.): Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory, Kluwer, Dordrecht, 1999.zbMATHGoogle Scholar
  30. 30.
    M. F. Janowitz: An order theoretic model for cluster analysis. SIAM J. App. Math. 34 (1978) 55–72.zbMATHGoogle Scholar
  31. 31.
    M. F. Janowitz: An ordinal model for cluster analysis - 15 years in retrospect, in From Data to Knowledge: Theoretical and Practical Aspects of Classification, Data Analysis and Knowledge Organization, ed. by W. Gaul, D. Pfeifer, Springer, Berlin, 1995, 58–72.Google Scholar
  32. 32.
    M. F. Janowitz, B. Schweizer: Ordinal and percentile clustering. Math. Social Sciences 18 (1989) 135–186.CrossRefMathSciNetzbMATHGoogle Scholar
  33. 33.
    H. Joe: Multivariate Models and Dependence Concepts, Chapman and Hall, London, 1997.CrossRefzbMATHGoogle Scholar
  34. 34.
    J. Kampé de Fériet: Le théorie généralisée de l’information et la mesure subjective de l’information Lecture Notes in Math. 398 (1974) 1–35.Google Scholar
  35. 35.
    J. Kampé de Fériet, B. Forte: Information et probabilité. C. R. Acad. Sci. Paris 265A (1967) 110–114; 142–146; 350–353.Google Scholar
  36. 36.
    X. Li, P. Mikusinski, H. Sherwood, M. D. Taylor: On approximation of copulas, in Distributions with Given Marginals and Moment Problems, ed. by V. Benei, J. Stépán, Kluwer, Dordrecht, 1997,106–116.Google Scholar
  37. 37.
    Gongfu Liao, Qinjie Fan: Minimal subshifts which display Schweizer-Smital chaos and have zero topological entropy. Science in China (Series A) 41 (1998) 33–41.CrossRefMathSciNetzbMATHGoogle Scholar
  38. 38.
    P. C. Mahalanobis: On the generalized distance in statistics. Proc. Nat. Inst. Sci. India 2 (1936) 49–55.zbMATHGoogle Scholar
  39. 39.
    K. Menger: Untersuchungen über allgemeine Metric I. Math. Ann. 100 (1928) 75–113.CrossRefMathSciNetzbMATHGoogle Scholar
  40. 40.
    K. Menger: Topology without points. The Rice Institute Pamphlet 27 (1940) 80–107.MathSciNetzbMATHGoogle Scholar
  41. 41.
    K. Menger: Statistical metrics. Proc. Nat. Acad. Sci. USA 28 (1942) 535–537.CrossRefMathSciNetzbMATHGoogle Scholar
  42. 42.
    K. Menger: The theory of relativity and geometry, in Albert Einstein, Philosopher-Scientist, Library of Living Philosophers, ed. by P. S. Schilpp, Evanston, IL, VII (1949) 459–474.Google Scholar
  43. 43.
    K. Menger: Probabilistic theories of relations. Proc. Nat. Acad. Sci. USA 37 (1951) 178–180.CrossRefMathSciNetzbMATHGoogle Scholar
  44. 44.
    K. Menger: Probabilistic geometry. Proc. Nat. Acad. Sci. USA 37 (1951) 226–229.CrossRefMathSciNetzbMATHGoogle Scholar
  45. 45.
    K. Menger: Ensembles flous et fonctions aléatoires. C. R. Acad. Sci. Paris 232 (1951) 2001–2003.MathSciNetzbMATHGoogle Scholar
  46. 46.
    K.Menger: Géométrie générale, Mem. Sci. Math. 124. Gauthier-Villars, Paris, 1954.Google Scholar
  47. 47.
    K. Menger: Random variables from the point of view of a general theory of variables, in Proc. Third Berkeley Symposium on Mathematical Statistics and Probability, ed. by L. M. LeCam, J. Neyman, Univ. of California Press, Berkeley and Los Angeles, 2 (1956) 215–229.Google Scholar
  48. 48.
    K. Menger: Mathematical implications of Mach’s ideas: Positivistic geometry, the clarification of functional connections, in Ernest Mach, Physicist and Philosopher, Boston Studies in the Philosophy of Science, ed. by R. S. Cohen, R. J. Seeger, Reidel, Dordrecht, 6 (1970) 107–125.Google Scholar
  49. 49.
    T. S. Motzkin: Sur le produit d’espaces métriques. C. R. Congres Int. Mathématiciens, Oslo, 2 (1936) 137–138.Google Scholar
  50. 50.
    R. Moynihan: On TT-semigroups of probability distribution functions II. Aeq. Math. 17 (1978) 19–40.MathSciNetzbMATHGoogle Scholar
  51. 51.
    R. Moynihan: Conjugate transforms and limit theorems for TT-semigroups. Studia Math. 69 (1980) 1–18.CrossRefMathSciNetzbMATHGoogle Scholar
  52. 52.
    R. B. Nelsen: Dependence and order in families of Archimedean copulas. J. Multivariate Anal. 60 (1997) 111–122.CrossRefMathSciNetzbMATHGoogle Scholar
  53. 53.
    R. B. Nelsen: An Introduction to Copulas, Lecture Notes in Statistics, v. 139, Springer, New York, 1999.Google Scholar
  54. 54.
    R. B. Nelsen, J. J. Quesada-Molina, B. Schweizer, C. Sempi: Derivability of some operations on distribution functions, in Distributions with Fixed Marginals and Related Topics, ed. by M. D. Taylor, B. Schweizer, L. Rüschendorf, IMS Lecture Notes - Monograph Series, 28 (1996) 233–243.CrossRefGoogle Scholar
  55. 55.
    H. T. Nguyen, E. Walker: A First Course in Fuzzy Logic. CRC Press, Boca Raton, 1997.zbMATHGoogle Scholar
  56. 56.
    R. C. Powers: Order automorphisms of spaces of nondecreasing functions. J. Math. Anal. Appl. 136 (1988) 112–123.CrossRefMathSciNetzbMATHGoogle Scholar
  57. 57.
    E. Prugovecki: Stochastic Quantum Mechanics and Quantum Spacetime, D. Reidel, Dordrecht, 1984.CrossRefzbMATHGoogle Scholar
  58. 58.
    M. Regenwetter, A. A. J. Marley: Random relations, random utilities and random functions. J. Math. Psychology 45 (2001) 864–912.CrossRefMathSciNetzbMATHGoogle Scholar
  59. 59.
    T. Riedel: Cauchy’s equation on A+. Aeq. Math. 41 (1991) 192–211.MathSciNetzbMATHGoogle Scholar
  60. 60.
    R. T. Rockafellar: Convex Analysis, Princeton University Press, 1970.Google Scholar
  61. 61.
    N. Rosen: Quantum geometry. Ann. Physics 19 (1962) 165–172.CrossRefMathSciNetzbMATHGoogle Scholar
  62. 62.
    B. Schweizer: Multiplications on the space of distribution functions. Aeq. Math. 12 (1975) 156–183.zbMATHGoogle Scholar
  63. 63.
    B. Schweizer: Thirty years of copulas, in Advances in Probability Distribution Functions with Given Marginals: Beyond the Copulas, ed. by G. Dall-Aglio, S. Kotz, G. Salinetti. Mathematics and its Applications, Kluwer, Dordrecht, 67 (1991) 13–50.Google Scholar
  64. 64.
    B. Schweizer: On the genesis of the notion of distributional chaos. Rendiconti del Seminario Matematico e Fisico di Milano 66 (1996) 159–167.CrossRefMathSciNetzbMATHGoogle Scholar
  65. 65.
    B. Schweizer, A. Sklar: Mesures aléatoires de l’information. C. R. Acad. Sci. Paris 269A (1969) 149–152.zbMATHGoogle Scholar
  66. 66.
    B. Schweizer, A. Sklar: Probabilistic Metric Spaces, Elsevier - North Holland, New York, 1983.zbMATHGoogle Scholar
  67. 67.
    B. Schweizer, A. Sklar, J. Smítal: Distributional (and other) chaos and its measurement. Real Anal. Exchange 27 (2001/2002) 495–524.MathSciNetzbMATHGoogle Scholar
  68. 68.
    B. Schweizer, J. Smítal: Measures of chaos and a spectral decomposition of dynamical systems on the interval. Trans. Amer. Math. Soc. 344 (1994) 737–754.CrossRefMathSciNetzbMATHGoogle Scholar
  69. 69.
    B. Schweizer, E. F. Wolff: On nonparametric measures of dependence of random variables. Ann. Statist. 9 (1981) 879–885.CrossRefMathSciNetzbMATHGoogle Scholar
  70. 70.
    A. N. Serstnev: On a probabilistic generalization of metric spaces. Kazan Gos. Univ. Ucen. Zap. 124 (1964) 3–11.MathSciNetGoogle Scholar
  71. 71.
    H. Sherwood: On E-spaces and their relation to other classes of probabilistic metric spaces. J. London Math. Soc. 44 (1969) 441–448.CrossRefMathSciNetzbMATHGoogle Scholar
  72. 72.
    H. Sherwood, M. D. Taylor: Doubly stochastic measures with hairpin support. Prob. Theory and Related Fields 78 (1988) 617–626.CrossRefMathSciNetzbMATHGoogle Scholar
  73. 73.
    A. Sklar: Fonctions de répartition à n dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris 8 (1959) 229–231.MathSciNetzbMATHGoogle Scholar
  74. 74.
    A. Spacek: Random metric spaces, Trans. Second Prague Conf. Information Theory, Decision Functions and Random Processes, Academic Press, New York, 1960, 627–638.Google Scholar
  75. 75.
    R. M. Tardiff: Topologies for probabilistic metric spaces. Pacific J. Math. 65 (1976) 233–251.CrossRefMathSciNetzbMATHGoogle Scholar
  76. 76.
    M. D. Taylor, B. Schweizer, L. Rüschendorf (eds.): Distributions with Fixed Marginals and Related Topics, IMS Lecture Notes — Monograph Series, v. 28 (1996).Google Scholar
  77. 77.
    E. Trillas, C. Alsina, L. Valverde: Do we need Max, Min and 1—j in fuzzy set theory?, in Fuzzy Set and Possibility Theory, Recent Developments, ed. by R. R. Yager, Pergamon Press, New York, 1982, 275–297.Google Scholar
  78. 78.
    A. Wald: On the Principles of Statistical Inference, Notre Dame Mathematical Lectures, No. 1 (1942).Google Scholar
  79. 79.
    A. Wald: On a statistical generalization of metric spaces. Proc. Nat. Acad. Sci. USA 29 (1943) 196–197.CrossRefMathSciNetzbMATHGoogle Scholar
  80. 80.
    L. Zadeh: Fuzzy sets. Information and Control 8 (1965) 338–353.CrossRefMathSciNetzbMATHGoogle Scholar
  81. 81.
    A. Sklar, J. Smítal: Distributional chaos on compact metric spaces via specification properties, J. Math. Anal. Appl. 241 (2000) 181–188.CrossRefMathSciNetzbMATHGoogle Scholar
  82. 82.
    E. P. Klement, R. Mesiar, E. Pap: Triangular Norms, Kluwer, Dordrecht, 2000.CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Wien 2003

Authors and Affiliations

  • B. Schweizer

There are no affiliations available

Personalised recommendations