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Central limit theorems for generalized Pólya urn models

Part of: Limit theorems Sequential methods

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.
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Abstract

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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
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.

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