Probability Theory and Related Fields

, Volume 74, Issue 2, pp 199–211 | Cite as

Best-possible bounds for the distribution of a sum — a problem of Kolmogorov

  • M. J. Frank
  • R. B. Nelsen
  • B. Schweizer


Recently, in answer to a question of Kolmogorov, G.D. Makarov obtained best-possible bounds for the distribution function of the sumX+Y of two random variables,X andY, whose individual distribution functions,FX andFY, are fixed. We show that these bounds follow directly from an inequality which has been known for some time. The techniques we employ, which are based on copulas and their properties, yield an insightful proof of the fact that these bounds are best-possible, settle the question of equality, and are computationally manageable. Furthermore, they extend to binary operations other than addition and to higher dimensions.


Distribution Function Stochastic Process Probability Theory High Dimension Mathematical Biology 
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.


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Copyright information

© Springer-Verlag 1987

Authors and Affiliations

  • M. J. Frank
    • 1
  • R. B. Nelsen
    • 2
  • B. Schweizer
    • 3
  1. 1.Department of MathematicsIllinois Institute of TechnologyChicagoUSA
  2. 2.Department of MathematicsLewis and Clark CollegePortlandUSA
  3. 3.Department of Mathematics and StatisticsUniversity of MassachusettsAmherstUSA

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