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Best-possible bounds for the distribution of a sum — a problem of Kolmogorov
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  • Published: June 1987

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

  • M. J. Frank1,
  • R. B. Nelsen2 &
  • B. Schweizer3 

Probability Theory and Related Fields volume 74, pages 199–211 (1987)Cite this article

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Summary

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,F X andF Y, 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.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Illinois Institute of Technology, 60616, Chicago, IL, USA

    M. J. Frank

  2. Department of Mathematics, Lewis and Clark College, 97219, Portland, OR, USA

    R. B. Nelsen

  3. Department of Mathematics and Statistics, University of Massachusetts, 01003, Amherst, MA, USA

    B. Schweizer

Authors
  1. M. J. Frank
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  2. R. B. Nelsen
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  3. B. Schweizer
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Cite this article

Frank, M.J., Nelsen, R.B. & Schweizer, B. Best-possible bounds for the distribution of a sum — a problem of Kolmogorov. Probab. Th. Rel. Fields 74, 199–211 (1987). https://doi.org/10.1007/BF00569989

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  • Received: 03 October 1984

  • Revised: 05 May 1986

  • Issue Date: June 1987

  • DOI: https://doi.org/10.1007/BF00569989

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Keywords

  • Distribution Function
  • Stochastic Process
  • Probability Theory
  • High Dimension
  • Mathematical Biology
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