Some characterizations of unimodal distribution functions

  • E. M. J. Bertin
  • W. Hengartner
  • R. Theodorescu


Let F be a distribution function and let QF(l)=0 for l<0 and QF(l)= sup {F(x+l)−F(x): x∈ℝ} for l≧0 be its Lévy concentration function. This paper has two purposes: to give a characterization of unimodal distribution functions (Theorem 3.5) and a representation theorem for the class of unimodal distribution functions (Theorem 6.2), both in terms of their Lévy concentration functions.


Distribution Function Stochastic Process Probability Theory Mathematical Biology Representation Theorem 
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  1. 1.
    Bourbaki, N.: Intégration. Chap. IX. Paris: Hermann. 1969Google Scholar
  2. 2.
    Hengartner, W., Theodorescu, R.: Concentration functions. New York: Academic Press, 1973; 2nd edition Moscow: Namca 1980Google Scholar
  3. 3.
    Hengartner, W., Theodorescu, R.: A characterization of strictly unimodal distribution functions by their concentration functions. Publ. Inst. Statist. Paris, 24, 1–10 (1978)Google Scholar
  4. 4.
    Gnedenko, B.V., Kolmogorov, A.N.: Limit distributions for sums of independent random variables. New York: Addison-Wesley, 1954Google Scholar
  5. 5.
    Lukacs, E.: Characteristic functions. 2nd. ed. New York: Hafner, 1970Google Scholar
  6. 6.
    Medgyessi, P.: Decomposition of superpositions of density functions and discrete distributions. Bristol: Adam Hilger, 1977Google Scholar
  7. 7.
    Rockafellar, R.T.: Convex analysis. Princeton: Princeton University Press, 1970Google Scholar
  8. 8.
    Smithson, R.E.: Multifunctions. Nieuw Archief voor Wiskunde (3), 20, 31–53 (1972)Google Scholar
  9. 9.
    Valentine, F.A.: Convex sets. New York: McGraw Hill, 1964Google Scholar

Copyright information

© Springer-Verlag 1981

Authors and Affiliations

  • E. M. J. Bertin
    • 1
  • W. Hengartner
    • 2
  • R. Theodorescu
    • 2
  1. 1.Department of MathematicsUniversity of UtrechtUtrechtNetherlands
  2. 2.Dept. of MathematicsLaval UniversityCanada

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