Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-24T10:57:07.277Z Has data issue: false hasContentIssue false
  • English
  • Français

Synchronization and fluctuation theorems for interacting Friedman urns

Part of: Special processes Limit theorems Probability theory on algebraic and topological structures

Published online by Cambridge University Press:  09 December 2016

Neeraja Sahasrabudhe
Neeraja Sahasrabudhe*
Affiliation:
Indian Institute of Technology Bombay
*
* Postal address:Indian Institute of Technology Bombay, Powai, Mumbai, 400076, Maharashtra, India. Email address: neeraja.budhey@iitb.ac.in
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

Keywords

Interacting urn modelFriedman urnPólya urnsynchronizationstable convergencereinforcementfluctuation theorem

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 60F05: Central limit and other weak theorems 60B10: Convergence of probability measures 60F99: None of the above, but in this section
Type
Research Papers
Information
Journal of Applied Probability , Volume 53 , Issue 4 , December 2016 , pp. 1221 - 1239
Copyright
Copyright © Applied Probability Trust 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Aldous, D. J. and Eagleson, G. K. (1978).On mixing and stability of limit theorems.Ann. Prob. 6,325–331.Google Scholar
[2] Benaïm, M.,Benjamini, I.,Chen, J. and Lima, Y. (2015).A generalized Pólya's urn with graph based interactions.Random Structures Algorithms 46,614–634.Google Scholar
[3] Berti, P.,Crimaldi, I.,Pratelli, L. and Rigo, P. (2010).Central limit theorems for multicolor urns with dominated colors.Stoch. Process. Appl. 120,1473–1491.Google Scholar
[4] Borkar, V. S. (2008).Stochastic Approximation: A Dynamical Systems Viewpoint.Cambridge University Press.CrossRefGoogle Scholar
[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.Google Scholar
[6] Crimaldi, I. (2009).An almost sure conditional convergence result and an application to a generalized Pólya urn.Internat. Math. Forum 4,1139–1156.Google Scholar
[7] Crimaldi, I.,Dai Pra, P. and Minelli, I. (2015).Fluctuation theorems for synchronization of interacting Pólya's urns.Stoch. Process. Appl. 126,930–947.Google Scholar
[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. 203–225.Google Scholar
[9] Dai Pra, P.,Louis, P.-Y. and Minelli, I. G. (2014).Synchronization via interacting reinforcement.J. Appl. Prob. 51,556–568.Google Scholar
[10] Feigin, P. D. (1985).Stable convergence of semimartingales.Stoch. Process. Appl. 19,125–134.Google Scholar
[11] Fisk, D. L. (1965).Quasi-martingales.Trans. Amer. Math. Soc. 120,369–389.Google Scholar
[12] Freedman, D. A. (1965).Bernard Friedman's urn.Ann. Math. Statist. 36,956–970.Google Scholar
[13] Friedman, B.(1949).A simple urn model.Commun. Pure Appl. Math. 2,59–70.CrossRefGoogle Scholar
[14] Hall, P. and Heyde, C. C. (1980).Martingale Limit Theory and Its Applications.Academic Press,New York.Google Scholar
[15] Laruelle, S. and Pagés, G. (2013).Randomized urn models revisited using stochastic approximation.Ann. Appl. Prob. 23,1409–1436.Google Scholar
[16] Launay, M. (2012).Interacting urn models.Preprint. Available at https://arxiv.org/abs/1101.1410.Google Scholar
[17] Mahmoud, H. (2009).Pólya Urn Models.CRC,Boca Raton, FL.Google Scholar
[18] Marsili, M. and Valleriani, A. (1998).Self organization of interacting Pólya urns.European Physical J. B 3,417–420.Google Scholar
[19] Paganoni, A. M. and Secchi, P. (2004).Interacting reinforced-urn systems.Adv. Appl. Prob. 36,791–804.Google Scholar
[20] Peccati, G. and Taqqu, M. S. (2008).Stable convergence of multiple Wiener‒Itô integrals.J. Theoret. Prob. 21,527–570.Google Scholar
[21] Pemantle, R. (2007).A survey of random processes with reinforcement.Prob. Surveys 4,1–79.Google Scholar
[22] Rao, K. M. (1969).Quasi-martingales.Math. Scand. 24,79–92.Google Scholar
[23] Rényi, A. (1963).On stable sequences of events.Sankhyā A 25,293–302.Google Scholar