Hostname: page-component-8448b6f56d-sxzjt Total loading time: 0 Render date: 2024-04-23T17:39:29.061Z Has data issue: false hasContentIssue false
  • English
  • Français

A central limit theorem, and related results, for a two-color randomly reinforced urn

Part of: Limit theorems

Published online by Cambridge University Press:  01 July 2016

Giacomo Aletti ,
Caterina May  and
Piercesare Secchi
Giacomo Aletti*
Affiliation:
Università degli Studi di Milano
Caterina May*
Affiliation:
Università del Piemonte Orientale
Piercesare Secchi*
Affiliation:
Politecnico di Milano
*
∗ Postal address: Dipartimento di Matematica ‘F. Enrigues’, Università degli Studi di Milano, Via Cesare Saldini 50, 20133 Milano, Italy. Email address: giacomo.aletti@mat.unimi.it
∗∗ Postal address: Dipartimento SEMEQ, Università del Piemonte Orientale, via Perrone 18, 28100 Novara, Italy. Email address: caterina.may@eco.unipmn.it
∗∗∗ Postal address: MOX – Modellistica e Calcolo Scientifico, Dipartimento di Matematica ‘F. Brioschi’, Politecnico di Milano, via Bonardi 9, 20133 Milano, Italy. Email address: piercesare.secchi@polimi.it
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove a central limit theorem for the sequence of random compositions of a two-color randomly reinforced urn. As a consequence, we are able to show that the distribution of the urn limit composition has no point masses.

Keywords

Reinforced processesgeneralized Pólya urnconvergence of conditional distributions

MSC classification

Primary: 60F05: Central limit and other weak theorems
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 41 , Issue 3 , September 2009 , pp. 829 - 844
Copyright
Copyright © Applied Probability Trust 2009 

References

Aletti, G., May, C. and Secchi, P. (2007). On the distribution of the limit proportion for a two-color, randomly reinforced urn with equal reinforcement distributions. Adv. Appl. Prob. 39, 690–707.CrossRefGoogle Scholar
Beggs, A. W. (2005). On the convergence of reinforcement learning. J. Econom. Theory 122, 1–36.CrossRefGoogle Scholar
Crimaldi, I. (2008). Almost sure conditional convergence for a generalized Pólya urn. Preprint, Dipartimento di Matematica, Università di Bologna. Available at: http://almadl.cib.unibo.it/.Google Scholar
Durham, S. D. and Yu, K. F. (1990). Randomized play-the leader rules for sequential sampling from two populations. Prob. Eng. Inf. Sci. 4, 355–367.CrossRefGoogle Scholar
Durham, S. D., Flournoy, N. and Li, W. (1998). A sequential design for maximizing the probability of a favourable response. Canad. J. Statist. 26, 479–495.CrossRefGoogle Scholar
Gibbs, A. L. and Su, F. E. (2002). On choosing and bounding probability metrics. Internat. Statist. Rev. 70, 419–435.CrossRefGoogle Scholar
Hopkins, E. and Posch, M. (2005). Attainability of boundary points under reinforcement learning. Games Econom. Behavior 53, 110–125.CrossRefGoogle Scholar
Li, W., Durham, S. D. and Flournoy, N. (1996). Randomized Pólya urn designs. In Proc. Biometric Section Amer. Statist. Assoc., American Statistical Association, Alexandria, VA, pp. 166–170.Google Scholar
May, C. and Flournoy, N. (2008). “Asymptotics in response-adaptive designs generated by a two-color, randomly reinforced urn. Ann. Statist. 32, 1058–1078.Google Scholar
May, C., Paganoni, A. and Secchi, P. (2005). On a two color generalized Pólya urn. Metron 63, 115–134.Google Scholar
May, C., Paganoni, A. and Secchi, P. (2007). Response-adaptive designs targeting the best treatment for clinical trials with continuous responses. In S. Co. 2007 5th Conf. - Complex Models and Computational Intensive Methods for Estimation and Prediction, Cluep, Padova, pp. 326–331.Google Scholar
Mera, M. E., Morán, M., Preiss, D. and Zajíček, L. (2003). Porosity, σ-porosity and measures. Nonlinearity 16, 247–255.CrossRefGoogle Scholar
Muliere, P., Paganoni, A. and Secchi, P. (2006). A randomly reinforced urn. J. Statist. Planning Infer. 136, 1853–1874.CrossRefGoogle Scholar
Paganoni, A. M. and Secchi, P. (2007). A numerical study for comparing two response-adaptive designs for continuous treatment effects. Statist. Meth. Appl. 16, 321–346.CrossRefGoogle Scholar
Pemantle, R. (1990). A time-dependent version of Pólya's urn. J. Theoret. Prob. 3, 627–637.CrossRefGoogle Scholar
Pemantle, R. and Volkov, S. (1999). Vertex-reinforced random walk on Z has finite range. Ann. Prob. 27, 1368–1388.CrossRefGoogle Scholar
Prokaj, V. (2001/02). On a construction of J. Tkadlec concerning σ-porous sets. Real Anal. Exchange 1, 269–273.CrossRefGoogle Scholar
Rackauskas, A. (1990). On probabilities of large deviations for martingales. Litovsk. Mat. Sb. 30, 784–795 (in Russian). English translation: Lithuanian Math. J. 30 (1991), 376-384.Google Scholar
Tkadlec, J. (1986/87). Construction of a finite Borel measure with σ-porous sets as null sets. Real Anal. Exchange 1, 349–353.CrossRefGoogle Scholar
Zajíček, L. (2005). On σ-porous sets in abstract spaces. Abstr. Appl. Anal. 5, 509–534.CrossRefGoogle Scholar