Skip to main content Accessibility help
Hostname: page-component-7f7b94f6bd-8p2w5 Total loading time: 0.258 Render date: 2022-06-28T15:02:20.545Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true } hasContentIssue true

Central limit theorems for a hypergeometric randomly reinforced urn

Published online by Cambridge University Press:  24 October 2016

Irene Crimaldi*
IMT School for Advanced Studies Lucca
* Postal address: IMT School for Advanced Studies Lucca, Piazza San Ponziano 6, 55100 Lucca, Italy. Email address:


We consider a variant of the randomly reinforced urn where more balls can be simultaneously drawn out and balls of different colors can be simultaneously added. More precisely, at each time-step, the conditional distribution of the number of extracted balls of a certain color, given the past, is assumed to be hypergeometric. We prove some central limit theorems in the sense of stable convergence and of almost sure conditional convergence, which are stronger than convergence in distribution. The proven results provide asymptotic confidence intervals for the limit proportion, whose distribution is generally unknown. Moreover, we also consider the case of more urns subjected to some random common factors.

Research Papers
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.)


[1] Aldous, D. J. and Eagleson, G. K. (1978).On mixing and stability of limit theorems.Ann. Prob. 6,325331.CrossRefGoogle Scholar
[2] Aletti, G.,May, C. and Secchi, P. (2009).A central limit theorem, and related results, for a two-color randomly reinforced urn.Adv. Appl. Prob. 41,829844.CrossRefGoogle Scholar
[3] Aoudia, D. A. and Perron, E. (2012).A new randomized Pólya urn model.Appl. Math. 3,21182122.CrossRefGoogle Scholar
[4] Arthur, W. B. (1989 ).Competing technologies, increasing returns, and lock-in by historical events.Econom. J. 99,!116131.Google Scholar
[5] Bai, Z.-D. and Hu, F. (2005).Asymptotics in randomized URN models.Ann. Appl. Prob. 15,914940.CrossRefGoogle Scholar
[6] Bai, Z. D.,Hu, F. and Rosenberger, W. F. (2002).Asymptotic properties of adaptive designs for clinical trials with delayed response.Ann. Statist. 30,122139.Google Scholar
[7] Banerjee, A.,Burlina, P. and Alajaji, F. (1999).Image segmentation and labeling using the Pólya urn model.IEEE Trans. Image Process. 8,12431253.CrossRefGoogle Scholar
[8] Bassetti, F.,Crimaldi, I. and Leisen, F. (2010).Conditionally identically distributed species sampling sequences.Adv. Appl. Prob. 42,433459.CrossRefGoogle Scholar
[9] Beggs, A. W. (2005).On the convergence of reinforcement learning.J. Econom. Theory 122,136.CrossRefGoogle Scholar
[10] Berti, P.,Pratelli, L. and Rigo, P. (2004).Limit theorems for a class of identically distributed random variables.Ann. Prob. 32,20292052.Google Scholar
[11] Berti, P.,crimaldi, I.,Pratelli, L. and Rigo, P. (2009).Rate of convergence of predictive distributions for dependent data.Bernoulli 15,13511367.CrossRefGoogle Scholar
[12] Berti, P.,crimaldi, I.,Pratelli, L. and Rigo, P. (2010).Central limit theorems for multicolor urns with dominated colors.Stoch. Process. Appl. 120,14731491.CrossRefGoogle Scholar
[13] Berti, P.,crimaldi, I.,Pratelli, L. and Rigo, P. (2011).A central limit theorem and its applications to multicolor randomly reinforced urns.J. Appl. Prob. 48,527546.CrossRefGoogle Scholar
[14] Berti, P.,crimaldi, I.,Pratelli, L. and Rigo, P. (2015).Central limit theorems for an Indian buffet model with random weights.Ann. Appl. Prob. 25,523547.CrossRefGoogle Scholar
[15] Blackwell, D. and Dubins, L. (1962).Merging of opinions with increasing information.Ann. Math. Statist. 33,882886.CrossRefGoogle Scholar
[16] Boldi, P.,Crimaldi, I. and Monti, C. (2016).A network model characterized by a latent attribute structure with competition.Inf. Sci. 354,236256.CrossRefGoogle Scholar
[17] Bose, A.,Dasgupta, A. and Maulik, K. (2009).Multicolor urn models with reducible replacement matrices.Bernoulli 15,279295.CrossRefGoogle Scholar
[18] Caldarelli, G.,Chessa, A.,Crimaldi, I. and Pammolli, F. (2013).Weighted networks as randomly reinforced urn processes.Phys. Rev. E 87,020106(R).CrossRefGoogle ScholarPubMed
[19] Chauvin, B.,Pouyanne, N. and Sahnoun, R. (2011).Limit distributions for large Pólya urns.Ann. Appl. Prob. 21,132.CrossRefGoogle Scholar
[20] Chen, M.-R. and Kuba, M. (2013).On generalized Pólya urn models.J. Appl. Prob. 50,11691186.CrossRefGoogle Scholar
[21] Chen, M.-R. and Wei, C.-Z. (2005).A new urn model.J. Appl. Prob. 42,964976.CrossRefGoogle Scholar
[22] Collevecchio, A.,Cotar, C. and LiCalzi, M. (2013).On a preferential attachment and generalized Póolya's urn model.Ann. Appl. Prob. 23,12191253.CrossRefGoogle Scholar
[23] Crimaldi, I. (2009).An almost sure conditional convergence result and an application to a generalized Pólya urn.Internat. Math. Forum 4,11391156.Google Scholar
[24] Crimaldi, I. and Leisen, F. (2008).Asymptotic results for a generalized Pólya urn with ‘multi-updating’ and applications to clinical trials.Commun. Statist. Theory Meth. 37,27772794.CrossRefGoogle Scholar
[25] Crimaldi, I.,Letta, G. and Pratelli, L. (2007).A strong form of stable convergence.Séminaire de Probabilités XL, (Lecture Notes Math. 1899}), Springer,Berlin, pp.203225.CrossRefGoogle Scholar
[26] Crimaldi, I. et al. (2016).Modeling networks with a growing feature-structure. Submitted.Google Scholar
[27] Dasgupta, A. and Maulik, K. (2011).Strong laws for urn models with balanced replacement matrices.Electron. J. Prob. 16,17231749.CrossRefGoogle Scholar
[28] Durham, S. D.,Flournoy, N. and Li, W. (1998).A sequential design for maximizing the probability of a favourable response..Canad. J. Statist. 26,479495.CrossRefGoogle Scholar
[29] Eggenberger, F. and Pólya, G. (1923).Über die statistik verketteter vorgänge.Z. Angew. Math. Mech. 3,279289.CrossRefGoogle Scholar
[30] Erev, I. and Roth, A. E. (1998).Predicting how people play games: reinforcement learning in experimental games with unique, mixed strategy equilibria.Amer. Econom. Rev. 88,848881.Google Scholar
[31] Feigin, P. D. (1985).Stable convergence of semimartingales.Stoch. Process. Appl. 19,125134.CrossRefGoogle Scholar
[32] Hall, P. and Heyde, C. C. (1980).Martingale Limit Theory and Its Application ,Academic Press ,New York.Google Scholar
[33] Hopkins, E. and Posch, M. (2005).Attainability of boundary points under reinforcement learning.Games Econom. Behav. 53,110125.CrossRefGoogle Scholar
[34] Hu, F. and Rosenberger, W. F. (2006).The Theory of Response-Adaptive Randomization in Clinical Trials.John Wiley,Hoboken, NJ.CrossRefGoogle Scholar
[35] Jacod, J. and Mémin, J. (1981).Sur un type de convergence intermédiaire entre la convergence en loi et la convergence en probabilité. InSéminaire de Probabilités XV (Lecture Notes Math. 850).Springer,Berlin, pp.529546.Google Scholar
[36] Janson, S. (2006).Limit theorems for triangular urn schemes.Prob. Theory Relat. Fields 134,417452.CrossRefGoogle Scholar
[37] Laruelle, S. and Pagès, G. (2013).Randomized urn models revisited using stochastic approximation.Ann. Appl. Prob. 23,14091436.CrossRefGoogle Scholar
[38] Mahmoud, H. M. (2009).Pólya Urn Models.CRC,Boca Raton, FL.Google Scholar
[39] Martin, C. F. and Ho, Y. C. (2002).Value of information in the Pólya urn process.Inf. Sci. 147,6590.CrossRefGoogle Scholar
[40] May, C. and Flournoy, N. (2009).Asymptotics in response-adaptive designs generated by a two-color, randomly reinforced urn.Ann. Statist. 37,10581078.CrossRefGoogle Scholar
[41] Muliere, P.,Paganoni, A. M. and Secchi, P. (2006).A randomly reinforced urn.J. Statist. Planning Inf. 136,18531874.CrossRefGoogle Scholar
[42] Peccati, G. and Taqqu, M. S. (2008).Stable convergence of multiple Wiener–Itô integrals.J. Theoret. Prob. 21,527570.CrossRefGoogle Scholar
[43] Pemantle, R. (2007).A survey of random processes with reinforcement.Prob. Surveys 4,179.CrossRefGoogle Scholar
[44] Pólya, G. (1930).Sur quelques points de la théorie des probabilités.Ann. Inst. H. Poincaré 1,117161.Google Scholar
[45] Rényi, A. (1963).On stable sequences of events.Sankhyā A 25,293302.Google Scholar
[46] Skyrms, B. and Pemantle, R. (2000). A dynamic model of social network formation.Proc. Nat. Acad. Sci. USA 97,93409346.CrossRefGoogle Scholar
[47] Xeu, J. (2006).A Pólya urn model of conformity.Working Paper 0614. Faculty of Economics, University of Cambridge.Google Scholar
[48] Zhang, L.-X. (2014).A Gaussian process approximation for two-color randomly reinforced urns.Electron. J. Prob. 19,86.CrossRefGoogle Scholar
[49] Zhang, L.-X.,Hu, F. and Cheung, S. H. (2006).Asymptotic theorems of sequential estimation-adjusted urn models.Ann. Appl. Prob. 16,340369.CrossRefGoogle Scholar
[50] Zhang, L.-X.,Hu, F.,Cheung, S. H.and Chan, W. S. (2014).Asymptotic properties of multicolor randomly reinforced Pólya urns.Adv. Appl. Prob. 46,585602.CrossRefGoogle Scholar
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the or variations. ‘’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Central limit theorems for a hypergeometric randomly reinforced urn
Available formats

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

Central limit theorems for a hypergeometric randomly reinforced urn
Available formats

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

Central limit theorems for a hypergeometric randomly reinforced urn
Available formats

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *