Global Dependency Measure for Sets of Random Elements: “The Italian Problem” and Some Consequences

  • Jordan Stoyanov
Part of the Trends in Mathematics book series (TM)


We suggest a detailed analysis of the classical independence/dependence properties for finite sets of random events or variables. All possible combinations of random elements are considered as a configuration obeying a hierarchical property. We define a function called a Dependency Measure taking values in the interval [0, 1] (0 corresponds to mutually independent sets; 1 corresponds to totally dependent sets) and serving as a global measure of the amount of dependency which is contained in the whole set of random elements. This leads to “The Italian Problem” about the existence of a probability model and a set of random elements with any prescribed independence/dependence structure. Some consequences and nonstandard illustrative examples are given. Related properties such as exchangeability and association are also discussed.


Random Event Random Vector Probability Space Dependency Measure Sample Space 
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. Bernstein, S. N. Theory of Probability. Gostechizdat, Moscow-Leningrad, 1928. [In Russian; previous lithographic edition 1916.]Google Scholar
  2. Block H. W., A. R. Sampson and T. H. Savits, eds. Topics in Statistical Dependence. IMS Ser., vol. 16, Inst. Math. Statist., Hayward (CA), 1991.Google Scholar
  3. Bohlmann, G. Die Grundbegriffe der Wahrscheinlichkeitsrechnung in Ihrer Anwendung auf die Lebensversicherung, pp. 244–278. In: ”Atti dei 4. Congresso Internationale del Matematici (Rome 1908),” ed. G. Castel-nouvo, vol. 3, Rome, 1908.Google Scholar
  4. Crow, E. L. A counterexample on independent events. Amer. Math. Monthly 74 (1967), 716–717.MathSciNetCrossRefGoogle Scholar
  5. Dall’Aglio G., S. Kotz and G. Salinetti, eds. Advances in Probability Distributions with Given Marginals (Proc. Symp. Rome’ 90), Kluwer Acad. Publ, Dordrecht, 1991.Google Scholar
  6. Eberlein, E. and M. Taqqu, eds. Dependence in Probability and Statistics. Birkhauser, Boston, 1986.zbMATHGoogle Scholar
  7. Mizera, I. and V. Balek. On the logical independence of the identities defining the stochastic independence of random events, Statist. & Probab. Letters 31 (1997), 281–284.MathSciNetzbMATHCrossRefGoogle Scholar
  8. Mori, T. F. and J. Stoyanov. Realizability of a probability model and random events with a prescribed independence/dependence structure, 1995/1996.Google Scholar
  9. Ruschendorf, L. Construction of multivariate distributions with given marginals. Annals Inst. Statist. Math. 37 (1986), 225–233.MathSciNetCrossRefGoogle Scholar
  10. Stoyanov, J. Counterexamples in Probability. John Wiley & Sons, Chichester-New York, 1987; 2nd ed. 1997.zbMATHGoogle Scholar
  11. Stoyanov, J. Dependency measure for sets of random events or random variables. Statist. & Probab. Letters 23 (1995), 108–115.MathSciNetGoogle Scholar
  12. Wang, Y. H. Dependent random variables with independent subsets-II. Canad. Math. Bull. 33 (1990), 24–28.MathSciNetzbMATHCrossRefGoogle Scholar
  13. Wang, Y. H. On the existence of a totally dependent probability space. J. Appl. Statist. Sci. 5 (1997), to appear.Google Scholar

Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • Jordan Stoyanov
    • 1
  1. 1.Instituto de MatematicsUniversidade Federal do Rio de JaneiroRio de JaneiroBrasil

Personalised recommendations