IDEAS home Printed from https://ideas.repec.org/a/spr/jotpro/v25y2012i4d10.1007_s10959-011-0399-7.html
   My bibliography  Save this article

A Functional Equation Whose Unknown is $\mathcal{P}([0,1])$ Valued

Author

Listed:
  • Giacomo Aletti

    (Università degli Studi di Milano)

  • Caterina May

    (Università del Piemonte Orientale)

  • Piercesare Secchi

    (Politecnico di Milano)

Abstract

We study a functional equation whose unknown maps a Euclidean space into the space of probability distributions on [0,1]. We prove existence and uniqueness of its solution under suitable regularity and boundary conditions, we show that it depends continuously on the boundary datum, and we characterize solutions that are diffuse on [0,1]. A canonical solution is obtained by means of a Randomly Reinforced Urn with different reinforcement distributions having equal means. The general solution to the functional equation defines a new parametric collection of distributions on [0,1] generalizing the Beta family.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jotpro:v:25:y:2012:i:4:d:10.1007_s10959-011-0399-7
    DOI: 10.1007/s10959-011-0399-7
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10959-011-0399-7
    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/s10959-011-0399-7?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. Anna Paganoni & Piercesare Secchi, 2007. "A numerical study for comparing two response-adaptive designs for continuous treatment effects," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 16(3), pages 321-346, November.
    2. Beggs, A.W., 2005. "On the convergence of reinforcement learning," Journal of Economic Theory, Elsevier, vol. 122(1), pages 1-36, May.
    3. 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. 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.
    2. Gerhold, Stefan & Gülüm, I. Cetin, 2019. "Peacocks nearby: Approximating sequences of measures," Stochastic Processes and their Applications, Elsevier, vol. 129(7), pages 2406-2436.
    3. Xuejun Zhao & Ruihao Zhu & William B. Haskell, 2022. "Learning to Price Supply Chain Contracts against a Learning Retailer," Papers 2211.04586, arXiv.org.
    4. 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).
    5. Marie Ernst & Yvik Swan, 2022. "Distances Between Distributions Via Stein’s Method," Journal of Theoretical Probability, Springer, vol. 35(2), pages 949-987, June.
    6. 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.
    7. Funai, Naoki, 2022. "Reinforcement learning with foregone payoff information in normal form games," Journal of Economic Behavior & Organization, Elsevier, vol. 200(C), pages 638-660.
    8. Naoki Funai, 2019. "Convergence results on stochastic adaptive learning," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 68(4), pages 907-934, November.
    9. Leandro Nascimento, 2022. "Bounded arbitrage and nearly rational behavior," Papers 2212.02680, arXiv.org, revised Jul 2023.
    10. Maxwell Pak & Bing Xu, 2016. "Generalized reinforcement learning in perfect-information games," International Journal of Game Theory, Springer;Game Theory Society, vol. 45(4), pages 985-1011, November.
    11. Patrick Marsh, 2019. "The role of information in nonstationary regression," Discussion Papers 19/04, University of Nottingham, Granger Centre for Time Series Econometrics.
    12. Jacques Durieu & Philippe Solal, 2012. "Models of Adaptive Learning in Game Theory," Chapters, in: Richard Arena & Agnès Festré & Nathalie Lazaric (ed.), Handbook of Knowledge and Economics, chapter 11, Edward Elgar Publishing.
    13. White, Staci A. & Herbei, Radu, 2015. "A Monte Carlo approach to quantifying model error in Bayesian parameter estimation," Computational Statistics & Data Analysis, Elsevier, vol. 83(C), pages 168-181.
    14. Laura Azzimonti & Francesca Ieva & Anna Maria Paganoni, 2013. "Nonlinear nonparametric mixed-effects models for unsupervised classification," Computational Statistics, Springer, vol. 28(4), pages 1549-1570, August.
    15. Mele, Antonio & Molnár, Krisztina & Santoro, Sergio, 2020. "On the perils of stabilizing prices when agents are learning," Journal of Monetary Economics, Elsevier, vol. 115(C), pages 339-353.
    16. Fabian Krüger & Sebastian Lerch & Thordis Thorarinsdottir & Tilmann Gneiting, 2021. "Predictive Inference Based on Markov Chain Monte Carlo Output," International Statistical Review, International Statistical Institute, vol. 89(2), pages 274-301, August.
    17. Hoang, Lê Nguyên & Soumis, François & Zaccour, Georges, 2019. "The return function: A new computable perspective on Bayesian–Nash equilibria," European Journal of Operational Research, Elsevier, vol. 279(2), pages 471-485.
    18. Stephan Eckstein & Gaoyue Guo & Tongseok Lim & Jan Obloj, 2019. "Robust pricing and hedging of options on multiple assets and its numerics," Papers 1909.03870, arXiv.org, revised Oct 2020.
    19. Nazaria Solferino & Viviana Solferino & Serena F. Taurino, 2018. "The economics analysis of a Q-learning model of cooperation with punishment and risk taking preferences," Journal of Economic Interaction and Coordination, Springer;Society for Economic Science with Heterogeneous Interacting Agents, vol. 13(3), pages 601-613, October.
    20. Berrendero, José R. & Cuevas, Antonio & Pateiro-López, Beatriz, 2016. "Shape classification based on interpoint distance distributions," Journal of Multivariate Analysis, Elsevier, vol. 146(C), pages 237-247.

    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:25:y:2012:i:4:d:10.1007_s10959-011-0399-7. 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.