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Synchronization and functional central limit theorems for interacting reinforced random walks

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  • Crimaldi, Irene
  • Dai Pra, Paolo
  • Louis, Pierre-Yves
  • Minelli, Ida G.

Abstract

We obtain Central Limit Theorems in Functional form for a class of time-inhomogeneous interacting random walks. Due to a reinforcement mechanism and interaction, the walks are strongly correlated and converge almost surely to the same, possibly random, limit. We study random walks interacting through a mean-field rule and compare the rate they converge to their limit with the rate of synchronization, i.e. the rate at which their mutual distances converge to zero. We show that, under certain conditions, synchronization is faster than convergence. Even if our focus is on theoretical results, we propose as main motivations two contexts in which such results could directly apply: urn models and opinion dynamics in a random network evolving via preferential attachment.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:129:y:2019:i:1:p:70-101
    DOI: 10.1016/j.spa.2018.02.012
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    References listed on IDEAS

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    1. Janson, Svante, 2004. "Functional limit theorems for multitype branching processes and generalized Pólya urns," Stochastic Processes and their Applications, Elsevier, vol. 110(2), pages 177-245, April.
    2. Pierluigi Contucci & Stefano Ghirlanda, 2007. "Modeling society with statistical mechanics: an application to cultural contact and immigration," Quality & Quantity: International Journal of Methodology, Springer, vol. 41(4), pages 569-578, August.
    3. Aletti, Giacomo & Ghiglietti, Andrea, 2017. "Interacting generalized Friedman’s urn systems," Stochastic Processes and their Applications, Elsevier, vol. 127(8), pages 2650-2678.
    4. 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.
    5. 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.
    6. 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.
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    Cited by:

    1. Yuan Hu & Abootaleb Shirvani & W. Brent Lindquist & Frank J. Fabozzi & Svetlozar T. Rachev, 2020. "Option Pricing Incorporating Factor Dynamics in Complete Markets," JRFM, MDPI, vol. 13(12), pages 1-33, December.
    2. Yuan Hu & Abootaleb Shirvani & W. Brent Lindquist & Frank J. Fabozzi & Svetlozar T. Rachev, 2020. "Option Pricing Incorporating Factor Dynamics in Complete Markets," Papers 2011.08343, arXiv.org.
    3. Irene Crimaldi & Pierre-Yves Louis & Ida Minelli, 2020. "Interacting non-linear reinforced stochastic processes: Synchronization and no-synchronization," Working Papers hal-02910341, HAL.
    4. Rosales, Rafael A. & Prado, Fernando P.A. & Pires, Benito, 2022. "Vertex reinforced random walks with exponential interaction on complete graphs," Stochastic Processes and their Applications, Elsevier, vol. 148(C), pages 353-379.

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