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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
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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. Beggs, A.W., 2005. "On the convergence of reinforcement learning," Journal of Economic Theory, Elsevier, vol. 122(1), pages 1-36, May.
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