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On the extreme points of moments sets

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  • Iosif Pinelis

    (Michigan Technological University)

Abstract

Necessary and sufficient conditions for a measure to be an extreme point of the set of measures on a given measurable space with prescribed generalized moments are given, as well as an application to extremal problems over such moment sets; these conditions are expressed in terms of atomic partitions of the measurable space. It is also shown that every such extreme measure can be adequately represented by a linear combination of k Dirac probability measures with nonnegative coefficients, where k is the number of restrictions on moments; moreover, when the measurable space has appropriate topological properties, the phrase “can be adequately represented by” here can be replaced simply by “is”. Applications to specific extremal problems are also given, including an exact lower bound on the exponential moments of truncated random variables, exact lower bounds on generalized moments of the interarrival distribution in queuing systems, and probability measures on product spaces with prescribed generalized marginal moments.

Suggested Citation

  • Iosif Pinelis, 2016. "On the extreme points of moments sets," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 83(3), pages 325-349, June.
    • Handle: RePEc:spr:mathme:v:83:y:2016:i:3:d:10.1007_s00186-015-0530-0
    DOI: 10.1007/s00186-015-0530-0
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    References listed on IDEAS

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    1. Ward Whitt, 1983. "Untold Horrors of the Waiting Room: What the Equilibrium Distribution Will Never Tell About the Queue-Length Process," Management Science, INFORMS, vol. 29(4), pages 395-408, April.
    2. Alan F. Karr, 1983. "Extreme Points of Certain Sets of Probability Measures, with Applications," Mathematics of Operations Research, INFORMS, vol. 8(1), pages 74-85, February.
    3. Gerhard Winkler, 1988. "Extreme Points of Moment Sets," Mathematics of Operations Research, INFORMS, vol. 13(4), pages 581-587, November.
    4. Tyner, Wally & Adams, John, 1976. "Rural Electrification In India: Biogas Versus Large Scale Power," 1976 Annual Meeting, August 15-18, State College, Pennsylvania 283822, American Agricultural Economics Association (New Name 2008: Agricultural and Applied Economics Association).
    5. A. E. Eckberg, 1977. "Sharp Bounds on Laplace-Stieltjes Transforms, with Applications to Various Queueing Problems," Mathematics of Operations Research, INFORMS, vol. 2(2), pages 135-142, May.
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    Cited by:

    1. Amparo Ba'illo & Javier C'arcamo & Carlos Mora-Corral, 2021. "Extremal points of Lorenz curves and applications to inequality analysis," Papers 2103.03286, arXiv.org.

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