IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v126y2016i3p930-947.html
   My bibliography  Save this article

Fluctuation theorems for synchronization of interacting Pólya’s urns

Author

Listed:
  • Crimaldi, Irene
  • Dai Pra, Paolo
  • Minelli, Ida Germana

Abstract

We consider a system of N two-colors urns in which the reinforcement of each urn depends also on the content of all the other urns. This interaction is of mean-field type and it is tuned by a parameter α∈[0,1]; in particular, for α=0 the N urns behave as N independent Pólya’s urns. For α>0 urns synchronize, in the sense that the fraction of balls of a given color converges a.s. to the same (random) limit in all urns. In this paper we study fluctuations around this synchronized regime. The scaling of these fluctuations depends on the parameter α. In particular the standard scaling t−1/2 appears only for α>1/2. For α≥1/2 we also determine the limit distribution of the rescaled fluctuations. We use the notion of stable convergence, which is stronger than convergence in distribution.

Suggested Citation

  • Crimaldi, Irene & Dai Pra, Paolo & Minelli, Ida Germana, 2016. "Fluctuation theorems for synchronization of interacting Pólya’s urns," Stochastic Processes and their Applications, Elsevier, vol. 126(3), pages 930-947.
    • Handle: RePEc:eee:spapps:v:126:y:2016:i:3:p:930-947
    DOI: 10.1016/j.spa.2015.10.005
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414915002537
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2015.10.005?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. Feigin, Paul D., 1985. "Stable convergence of semimartingales," Stochastic Processes and their Applications, Elsevier, vol. 19(1), pages 125-134, February.
    2. Berti, Patrizia & Crimaldi, Irene & Pratelli, Luca & Rigo, Pietro, 2010. "Central limit theorems for multicolor urns with dominated colors," Stochastic Processes and their Applications, Elsevier, vol. 120(8), pages 1473-1491, August.
    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.
    4. M. Marsili & A. Valleriani, 1998. "Self organization of interacting polya urns," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 3(4), pages 417-420, June.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Aletti, Giacomo & Ghiglietti, Andrea, 2017. "Interacting generalized Friedman’s urn systems," Stochastic Processes and their Applications, Elsevier, vol. 127(8), pages 2650-2678.
    2. Emilien Macault, 2022. "Stochastic Consensus and the Shadow of Doubt," Papers 2201.12100, arXiv.org.
    3. Crimaldi, Irene & Louis, Pierre-Yves & Minelli, Ida G., 2022. "An urn model with random multiple drawing and random addition," Stochastic Processes and their Applications, Elsevier, vol. 147(C), pages 270-299.
    4. 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.
    5. Crimaldi, Irene & Louis, Pierre-Yves & Minelli, Ida G., 2023. "Statistical test for an urn model with random multidrawing and random addition," Stochastic Processes and their Applications, Elsevier, vol. 158(C), pages 342-360.

    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. 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.
    2. 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.
    3. Gerhold, Stefan & Gülüm, I. Cetin, 2019. "Peacocks nearby: Approximating sequences of measures," Stochastic Processes and their Applications, Elsevier, vol. 129(7), pages 2406-2436.
    4. Xuejun Zhao & Ruihao Zhu & William B. Haskell, 2022. "Learning to Price Supply Chain Contracts against a Learning Retailer," Papers 2211.04586, arXiv.org.
    5. 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).
    6. Marie Ernst & Yvik Swan, 2022. "Distances Between Distributions Via Stein’s Method," Journal of Theoretical Probability, Springer, vol. 35(2), pages 949-987, June.
    7. Küchler, Uwe & Sørensen, Michael M., 1998. "A note on limit theorems for multivariate martingales," SFB 373 Discussion Papers 1998,45, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
    8. Leandro Nascimento, 2022. "Bounded arbitrage and nearly rational behavior," Papers 2212.02680, arXiv.org, revised Jul 2023.
    9. 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.
    10. Patrick Marsh, 2019. "The role of information in nonstationary regression," Discussion Papers 19/04, University of Nottingham, Granger Centre for Time Series Econometrics.
    11. 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.
    12. 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.
    13. 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.
    14. 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.
    15. 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.
    16. 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.
    17. Patrick Marsh, "undated". "A Measure of Distance for the Unit Root Hypothesis," Discussion Papers 05/02, Department of Economics, University of York.
    18. Teo Sharia, 2010. "Recursive parameter estimation: asymptotic expansion," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 62(2), pages 343-362, April.
    19. Crimaldi, Irene & Louis, Pierre-Yves & Minelli, Ida G., 2022. "An urn model with random multiple drawing and random addition," Stochastic Processes and their Applications, Elsevier, vol. 147(C), pages 270-299.
    20. Zellinger, Werner & Moser, Bernhard A., 2021. "On the truncated Hausdorff moment problem under Sobolev regularity conditions," Applied Mathematics and Computation, Elsevier, vol. 400(C).

    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:eee:spapps:v:126:y:2016:i:3:p:930-947. 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: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    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.