Hostname: page-component-74656f9884-j7cmm Total loading time: 0 Render date: 2026-11-10T05:18:03.813Z Has data issue: false hasContentIssue false
  • English
  • Français

Central limit theorems for generalized Pólya urn models

Part of: Sequential methods Limit theorems

Published online by Cambridge University Press:  14 July 2016

I. Higueras ,
J. Moler ,
F. Plo  and
M. San Miguel
I. Higueras*
Affiliation:
Universidad Pública de Navarra
J. Moler*
Affiliation:
Universidad Pública de Navarra
F. Plo*
Affiliation:
Universidad de Zaragoza
M. San Miguel*
Affiliation:
Universidad de Zaragoza
*
∗ Postal address: Departamento de Matemática e Informática, Universidad Pública de Navarra, Campus de Arrosadía, 31006 Pamplona, Spain.
∗∗ Postal address: Departamento de Estadística e Investigación Operativa, Universidad Pública de Navarra, Campus de Arrosadía, 31006 Pamplona, Spain. Email address: jmoler@unavarra.es
∗∗∗ Postal address: Departamento de Métodos Estadísticos, Universidad de Zaragoza, Pedro Cerbuna 12, 50009 Zaragoza, Spain.
∗∗∗ Postal address: Departamento de Métodos Estadísticos, Universidad de Zaragoza, Pedro Cerbuna 12, 50009 Zaragoza, Spain.
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.

In this paper we obtain central limit theorems for generalized Pólya urn models with L ≥ 2 colors where one out of K different replacements (actions) is applied randomly at each step. Each possible action constitutes a row of the replacement matrix, which can be nonsquare and random. The actions are chosen following a probability distribution given by an arbitrary function of the proportions of the balls of different colors present in the urn. Moreover, under the same hypotheses it is proved that the covariance matrix of the asymptotic distribution is the solution of a Lyapunov equation, and a procedure is given to obtain the covariance matrix in an explicit form. Some applications of these results to random trees and adaptive designs in clinical trials are also presented.

Keywords

Generalized Pólya urn modelcentral limit theoremRobbins-Monro recurrence equation

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 62L20: Stochastic approximation

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 43 , Issue 4 , December 2006 , pp. 938 - 951
Copyright
© Applied Probability Trust 2006 

Footnotes

Partially supported by DGA project E22 and MEC project MTM2004-01175.

References

Artur, B., Ermol′ev, Yu. M. and Kaniovskij, Yu. M. (1983). {A generalized urn problem and its applications.} Cybernetics {19,} 61–71.CrossRefGoogle Scholar
Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, Berlin.CrossRefGoogle Scholar
Bai, Z. D. and Hu, F. (1999). {Asymptotic theorems for urn models with nonhomogeneous generating matrices}. Stoch. Process. Appl. {80,} 87–101.CrossRefGoogle Scholar
Bai, Z. D. and Hu, F. (2005). {Asymptotics in randomized urn models}. Ann. Appl. Prob. {15,} 914–940.Google Scholar
Duflo, M. (1996). Algorithmes Stochastiques. Springer, Paris.Google Scholar
Gouet, R. (1993). {Martingale functional central limit theorems for a generalized Pólya urn}. Ann. Prob. {21,} 1624–1639.Google Scholar
Higueras, I., Moler, J., Plo, F. and San Miguel, M. (2003). Urn models and differential algebraic equations. J. Appl. Prob. 40, 401–412.CrossRefGoogle Scholar
Hill, B. M., Lane, D. and Sudderth, W. (1980). {A strong law for some generalized urn processes}. Ann. Prob. {8,} 214–226.Google Scholar
Janson, S. (2004). Functional limit theorems for multitype branching processes and generalized Pólya urns. Stoch. Process. Appl. 110, 177–245.Google Scholar
Khalil, H. K. (1996). Nonlinear Systems, 2nd edn. Prentice-Hall, Upper Saddle River, NJ.Google Scholar
Kushner, H. J. and Yin, G. G. (1997). Stochastic Approximation Algorithms and Applications (Appl. Math. 35). Springer, Berlin.Google Scholar
Mahmoud, H. M. (2002). The size of random bucket trees via urn models. Acta Informatica 38, 813–838.Google Scholar
Mahmoud, H. M. and Smythe, R. T. (1995). Probabilistic analysis of bucket recursive trees. Theoret. Comput. Sci. 144, 221–249.CrossRefGoogle Scholar
Matthews, P. C. and Rosenberger, W. F. (1997). Variance in randomized play-the-winner clinical trials. Statist. Prob. Lett. 35, 233–240.CrossRefGoogle Scholar
Moler, J. A., Plo, F. and San Miguel, M. (2005). Adaptive designs and Robbins–Monro algorithm. J. Statist. Planning Infer. 131, 161–174.Google Scholar
Moler, J. A., Plo, F. and San Miguel, M. (2006). An adaptive design for clinical trials with non-dichotomous response and prognostic factors. Statist. Prob. Lett. 76, 1940–1940.Google Scholar
Pelletier, M. (1998). Weak convergence rates for stochastic approximation with application to multiple targets and simulated annealing. Ann. Appl. Prob. 8, 10–44.Google Scholar
Rosenberger, W. F. (2002). Randomized urn models and sequential designs. Sequent. Anal. 21, 1–28.Google Scholar
Seneta, E. (1981). Non-Negative Matrices and Markov Chains, 2nd edn. Springer, New York.Google Scholar
Smythe, R. T. (1996). {Central limit theorems for urn models.} Stoch. Process. Appl. {65,} 115–137.Google Scholar
Smythe, R. T. and Mahmoud, H. M. (1995). A survey of recursive trees. Theory Prob. Math. Statist. 51, 1–27.Google Scholar
Tsukiji, T. and Mahmoud, H. M. (2001). A limit law for outputs in random recursive circuits. Algorithmica 31, 403–412.Google Scholar
Wei, L. J. and Durham, S. (1978). The randomized play-the-winner rule in medical trials. J. Amer. Statist. Soc. {73,} 840–843.Google Scholar