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Fluctuation theorems for synchronization of interacting Pólya’s urns

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  • 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
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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. 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.
    4. Feigin, Paul D., 1985. "Stable convergence of semimartingales," Stochastic Processes and their Applications, Elsevier, vol. 19(1), pages 125-134, February.
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    Cited by:

    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. Aletti, Giacomo & Ghiglietti, Andrea, 2017. "Interacting generalized Friedman’s urn systems," Stochastic Processes and their Applications, Elsevier, vol. 127(8), pages 2650-2678.
    3. 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.
    4. Emilien Macault, 2022. "Stochastic Consensus and the Shadow of Doubt," Papers 2201.12100, arXiv.org.
    5. 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.

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