IDEAS home Printed from https://ideas.repec.org/a/spr/jotpro/v23y2010i2d10.1007_s10959-009-0216-8.html
   My bibliography  Save this article

Stochastic Relations of Random Variables and Processes

Author

Listed:
  • Lasse Leskelä

    (Helsinki University of Technology)

Abstract

This paper generalizes the notion of stochastic order to a relation between probability measures over arbitrary measurable spaces. This generalization is motivated by the observation that for the stochastic ordering of two stationary Markov processes, it suffices that the generators of the processes preserve some, not necessarily reflexive or transitive, subrelation of the order relation. The main contributions of the paper are: a functional characterization of stochastic relations, necessary and sufficient conditions for the preservation of stochastic relations, and an algorithm for finding subrelations preserved by probability kernels. The theory is illustrated with applications to hidden Markov processes, population processes, and queueing systems.

Suggested Citation

  • Lasse Leskelä, 2010. "Stochastic Relations of Random Variables and Processes," Journal of Theoretical Probability, Springer, vol. 23(2), pages 523-546, June.
    • Handle: RePEc:spr:jotpro:v:23:y:2010:i:2:d:10.1007_s10959-009-0216-8
    DOI: 10.1007/s10959-009-0216-8
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10959-009-0216-8
    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/s10959-009-0216-8?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. William A. Massey, 1987. "Stochastic Orderings for Markov Processes on Partially Ordered Spaces," Mathematics of Operations Research, INFORMS, vol. 12(2), pages 350-367, May.
    2. Ward Whitt, 1986. "Stochastic Comparisons for Non-Markov Processes," Mathematics of Operations Research, INFORMS, vol. 11(4), pages 608-618, November.
    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. Dubra, Juan & Maccheroni, Fabio & Ok, Efe A., 2004. "Expected utility theory without the completeness axiom," Journal of Economic Theory, Elsevier, vol. 115(1), pages 118-133, March.
    2. Denuit, Michel & Vylder, Etienne De & Lefevre, Claude, 1999. "Extremal generators and extremal distributions for the continuous s-convex stochastic orderings," Insurance: Mathematics and Economics, Elsevier, vol. 24(3), pages 201-217, May.
    3. Xuefeng Gao & S. J. Deng, 2014. "Hydrodynamic limit of order book dynamics," Papers 1411.7502, arXiv.org, revised Feb 2016.
    4. William A. Massey & Emmanuel Ekwedike & Robert C. Hampshire & Jamol J. Pender, 2023. "A transient symmetry analysis for the M/M/1/k queue," Queueing Systems: Theory and Applications, Springer, vol. 103(1), pages 1-43, February.
    5. Okada, Daijiro & Tercieux, Olivier, 2012. "Log-linear dynamics and local potential," Journal of Economic Theory, Elsevier, vol. 147(3), pages 1140-1164.
    6. Christoph Heinzel, 2014. "Term structure of discount rates under multivariate s-ordered consumption growth," Working Papers SMART 14-01, INRAE UMR SMART.
    7. Dijk, N.M. van, 1989. "Sensitivity error bounds for non-exponental stochastic networks," Serie Research Memoranda 0093, VU University Amsterdam, Faculty of Economics, Business Administration and Econometrics.
    8. Dai Pra, Paolo & Louis, Pierre-Yves & Minelli, Ida Germana, 2010. "Realizable monotonicity for continuous-time Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 120(6), pages 959-982, June.
    9. Pekergin, Nihal & Dayar, Tugrul & Alparslan, Denizhan N., 2005. "Componentwise bounds for nearly completely decomposable Markov chains using stochastic comparison and reordering," European Journal of Operational Research, Elsevier, vol. 165(3), pages 810-825, September.
    10. Forbes, Florence & François, Olivier, 1997. "Stochastic comparison for Markov processes on a product of partially ordered sets," Statistics & Probability Letters, Elsevier, vol. 33(3), pages 309-320, May.
    11. Frostig, Esther & Denuit, Michel, 2006. "Monotonicity results for portfolios with heterogeneous claims arrival processes," Insurance: Mathematics and Economics, Elsevier, vol. 38(3), pages 484-494, June.
    12. Lasse Leskelä, 2022. "Ross’s second conjecture and supermodular stochastic ordering," Queueing Systems: Theory and Applications, Springer, vol. 100(3), pages 213-215, April.
    13. Chuancun Yin, 2019. "Stochastic Orderings of Multivariate Elliptical Distributions," Papers 1910.07158, arXiv.org, revised Nov 2019.
    14. Dominik Karos, 2016. "Coordinated Adoption of Social Innovations," Economics Series Working Papers 797, University of Oxford, Department of Economics.
    15. Denuit, Michel, 2000. "Time stochastic s-convexity of claim processes," Insurance: Mathematics and Economics, Elsevier, vol. 26(2-3), pages 203-211, May.
    16. Dijk, N.M. van, 1989. "The importance of bias-terms for error bounds and comparison results," Serie Research Memoranda 0036, VU University Amsterdam, Faculty of Economics, Business Administration and Econometrics.
    17. Ferreira, Fátima & Pacheco, António, 2007. "Comparison of level-crossing times for Markov and semi-Markov processes," Statistics & Probability Letters, Elsevier, vol. 77(2), pages 151-157, January.
    18. Jing‐Sheng Song & Paul H. Zipkin, 2012. "Newsvendor problems with sequentially revealed demand information," Naval Research Logistics (NRL), John Wiley & Sons, vol. 59(8), pages 601-612, December.
    19. Bar Light, 2019. "Stochastic Comparative Statics in Markov Decision Processes," Papers 1904.05481, arXiv.org, revised Jan 2020.
    20. van Dijk, Nico M., 2008. "Error bounds for state space truncation of finite Jackson networks," European Journal of Operational Research, Elsevier, vol. 186(1), pages 164-181, April.

    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:23:y:2010:i:2:d:10.1007_s10959-009-0216-8. 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.