IDEAS home Printed from https://ideas.repec.org/a/spr/jotpro/v33y2020i2d10.1007_s10959-019-00893-1.html
   My bibliography  Save this article

Asymptotics of One-Dimensional Lévy Approximations

Author

Listed:
  • Arno Berger

    (University of Alberta)

  • Chuang Xu

    (University of Copenhagen)

Abstract

For arbitrary Borel probability measures on the real line, necessary and sufficient conditions are presented that characterize best purely atomic approximations relative to the classical Lévy probability metric, given any number of atoms, and allowing for additional constraints regarding locations or weights of atoms. The precise asymptotics (as the number of atoms goes to infinity) of the approximation error is identified for the important special cases of best uniform (i.e. all atoms having equal weight) and best (i.e. unconstrained) approximations, respectively. When compared to similar results known for other probability metrics, the results for Lévy approximations are more complete and require fewer assumptions.

Suggested Citation

  • Arno Berger & Chuang Xu, 2020. "Asymptotics of One-Dimensional Lévy Approximations," Journal of Theoretical Probability, Springer, vol. 33(2), pages 1164-1195, June.
    • Handle: RePEc:spr:jotpro:v:33:y:2020:i:2:d:10.1007_s10959-019-00893-1
    DOI: 10.1007/s10959-019-00893-1
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10959-019-00893-1
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10959-019-00893-1?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Arno Berger & Theodore P. Hill & Kent E. Morrison, 2008. "Scale-Distortion Inequalities for Mantissas of Finite Data Sets," Journal of Theoretical Probability, Springer, vol. 21(1), pages 97-117, March.
    2. S. Dereich & C. Vormoor, 2011. "The High Resolution Vector Quantization Problem with Orlicz Norm Distortion," Journal of Theoretical Probability, Springer, vol. 24(2), pages 517-544, June.
    3. Georg Ch. Pflug & Alois Pichler, 2011. "Approximations for Probability Distributions and Stochastic Optimization Problems," International Series in Operations Research & Management Science, in: Marida Bertocchi & Giorgio Consigli & Michael A. H. Dempster (ed.), Stochastic Optimization Methods in Finance and Energy, edition 1, chapter 0, pages 343-387, Springer.
    4. Alison L. Gibbs & Francis Edward Su, 2002. "On Choosing and Bounding Probability Metrics," International Statistical Review, International Statistical Institute, vol. 70(3), pages 419-435, December.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Arno Berger & Chuang Xu, 2019. "Best Finite Approximations of Benford’s Law," Journal of Theoretical Probability, Springer, vol. 32(3), pages 1525-1553, September.
    2. Yannan Chen & Hailin Sun & Huifu Xu, 2021. "Decomposition and discrete approximation methods for solving two-stage distributionally robust optimization problems," Computational Optimization and Applications, Springer, vol. 78(1), pages 205-238, January.
    3. Yongchao Liu & Alois Pichler & Huifu Xu, 2019. "Discrete Approximation and Quantification in Distributionally Robust Optimization," Mathematics of Operations Research, INFORMS, vol. 44(1), pages 19-37, February.
    4. Wei Wang & Huifu Xu & Tiejun Ma, 2020. "Quantitative Statistical Robustness for Tail-Dependent Law Invariant Risk Measures," Papers 2006.15491, arXiv.org.
    5. Fenner, Trevor & Levene, Mark & Loizou, George, 2010. "Predicting the long tail of book sales: Unearthing the power-law exponent," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(12), pages 2416-2421.
    6. Bomze, Immanuel M. & Gabl, Markus & Maggioni, Francesca & Pflug, Georg Ch., 2022. "Two-stage stochastic standard quadratic optimization," European Journal of Operational Research, Elsevier, vol. 299(1), pages 21-34.
    7. Gerhold, Stefan & Gülüm, I. Cetin, 2019. "Peacocks nearby: Approximating sequences of measures," Stochastic Processes and their Applications, Elsevier, vol. 129(7), pages 2406-2436.
    8. Xuejun Zhao & Ruihao Zhu & William B. Haskell, 2022. "Learning to Price Supply Chain Contracts against a Learning Retailer," Papers 2211.04586, arXiv.org.
    9. Gurdip Bakshi & Xiaohui Gao & George Panayotov, 2021. "A Theory of Dissimilarity Between Stochastic Discount Factors," Management Science, INFORMS, vol. 67(7), pages 4602-4622, July.
    10. Puppo, L. & Pedroni, N. & Maio, F. Di & Bersano, A. & Bertani, C. & Zio, E., 2021. "A Framework based on Finite Mixture Models and Adaptive Kriging for Characterizing Non-Smooth and Multimodal Failure Regions in a Nuclear Passive Safety System," Reliability Engineering and System Safety, Elsevier, vol. 216(C).
    11. Marie Ernst & Yvik Swan, 2022. "Distances Between Distributions Via Stein’s Method," Journal of Theoretical Probability, Springer, vol. 35(2), pages 949-987, June.
    12. Crimaldi, Irene & Dai Pra, Paolo & Louis, Pierre-Yves & Minelli, Ida G., 2019. "Synchronization and functional central limit theorems for interacting reinforced random walks," Stochastic Processes and their Applications, Elsevier, vol. 129(1), pages 70-101.
    13. Pesenti, Silvana M. & Millossovich, Pietro & Tsanakas, Andreas, 2019. "Reverse sensitivity testing: What does it take to break the model?," European Journal of Operational Research, Elsevier, vol. 274(2), pages 654-670.
    14. Moritz Nobis & Carlo Schmitt & Ralf Schemm & Armin Schnettler, 2020. "Pan-European CVaR-Constrained Stochastic Unit Commitment in Day-Ahead and Intraday Electricity Markets," Energies, MDPI, vol. 13(9), pages 1-35, May.
    15. Walter Gutjahr & Alois Pichler, 2016. "Stochastic multi-objective optimization: a survey on non-scalarizing methods," Annals of Operations Research, Springer, vol. 236(2), pages 475-499, January.
    16. Matheus Henrique Junqueira Saldanha & Adriano Kamimura Suzuki, 2023. "On dealing with the unknown population minimum in parametric inference," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 107(3), pages 509-535, September.
    17. Walter J. Gutjahr & Alois Pichler, 2016. "Stochastic multi-objective optimization: a survey on non-scalarizing methods," Annals of Operations Research, Springer, vol. 236(2), pages 475-499, January.
    18. Beltran-Royo, C., 2017. "Two-stage stochastic mixed-integer linear programming: The conditional scenario approach," Omega, Elsevier, vol. 70(C), pages 31-42.
    19. Leandro Nascimento, 2022. "Bounded arbitrage and nearly rational behavior," Papers 2212.02680, arXiv.org, revised Jul 2023.
    20. Giacomo Aletti & Caterina May & Piercesare Secchi, 2012. "A Functional Equation Whose Unknown is $\mathcal{P}([0,1])$ Valued," Journal of Theoretical Probability, Springer, vol. 25(4), pages 1207-1232, December.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:jotpro:v:33:y:2020:i:2:d:10.1007_s10959-019-00893-1. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.