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Synchronization and fluctuation theorems for interacting Friedman urns

  • Neeraja Sahasrabudhe (a1)


We consider a model of N interacting two-colour Friedman urns. The interaction model considered is such that the reinforcement of each urn depends on the fraction of balls of a particular colour in that urn as well as the overall fraction of balls of that colour in all the urns combined together. We show that the urns synchronize almost surely and that the fraction of balls of each colour converges to the deterministic limit of one-half, which matches with the limit known for a single Friedman urn. Furthermore, we use the notion of stable convergence to obtain limit theorems for fluctuations around the synchronization limit.


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* Postal address:Indian Institute of Technology Bombay, Powai, Mumbai, 400076, Maharashtra, India. Email address:


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[1] Aldous, D. J. and Eagleson, G. K. (1978).On mixing and stability of limit theorems.Ann. Prob. 6,325331.
[2] Benaïm, M.,Benjamini, I.,Chen, J. and Lima, Y. (2015).A generalized Pólya's urn with graph based interactions.Random Structures Algorithms 46,614634.
[3] Berti, P.,Crimaldi, I.,Pratelli, L. and Rigo, P. (2010).Central limit theorems for multicolor urns with dominated colors.Stoch. Process. Appl. 120,14731491.
[4] Borkar, V. S. (2008).Stochastic Approximation: A Dynamical Systems Viewpoint.Cambridge University Press.
[5] Cirillo, P.,Gallegati, M. and Hüsler, J. (2012).A Pólya lattice model to study leverage dynamics and contagious financial fragility.Adv. Complex Systems 15,1250069.
[6] Crimaldi, I. (2009).An almost sure conditional convergence result and an application to a generalized Pólya urn.Internat. Math. Forum 4,11391156.
[7] Crimaldi, I.,Dai Pra, P. and Minelli, I. (2015).Fluctuation theorems for synchronization of interacting Pólya's urns.Stoch. Process. Appl. 126,930947.
[8] Crimaldi, I.,Letta, G. and Pratelli, L. (2007).A strong form of stable convergence.In Séminaire de Probabilités XL(Lecture Notes Math. 1899),Springer,Berlin,pp. 203225.
[9] Dai Pra, P.,Louis, P.-Y. and Minelli, I. G. (2014).Synchronization via interacting reinforcement.J. Appl. Prob. 51,556568.
[10] Feigin, P. D. (1985).Stable convergence of semimartingales.Stoch. Process. Appl. 19,125134.
[11] Fisk, D. L. (1965).Quasi-martingales.Trans. Amer. Math. Soc. 120,369389.
[12] Freedman, D. A. (1965).Bernard Friedman's urn.Ann. Math. Statist. 36,956970.
[13] Friedman, B.(1949).A simple urn model.Commun. Pure Appl. Math. 2,5970.
[14] Hall, P. and Heyde, C. C. (1980).Martingale Limit Theory and Its Applications.Academic Press,New York.
[15] Laruelle, S. and Pagés, G. (2013).Randomized urn models revisited using stochastic approximation.Ann. Appl. Prob. 23,14091436.
[16] Launay, M. (2012).Interacting urn models.Preprint. Available at
[17] Mahmoud, H. (2009).Pólya Urn Models.CRC,Boca Raton, FL.
[18] Marsili, M. and Valleriani, A. (1998).Self organization of interacting Pólya urns.European Physical J. B 3,417420.
[19] Paganoni, A. M. and Secchi, P. (2004).Interacting reinforced-urn systems.Adv. Appl. Prob. 36,791804.
[20] Peccati, G. and Taqqu, M. S. (2008).Stable convergence of multiple Wiener‒Itô integrals.J. Theoret. Prob. 21,527570.
[21] Pemantle, R. (2007).A survey of random processes with reinforcement.Prob. Surveys 4,179.
[22] Rao, K. M. (1969).Quasi-martingales.Math. Scand. 24,7992.
[23] Rényi, A. (1963).On stable sequences of events.Sankhyā A 25,293302.


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Synchronization and fluctuation theorems for interacting Friedman urns

  • Neeraja Sahasrabudhe (a1)


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