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Urn models and differential algebraic equations

Part of: Limit theorems Stability theory

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, Campus Arrosadía, 31015 Pamplona, Spain.
∗∗ Postal address: Departamento de Estadística e Investigación Operativa, Campus Arrosadía, 31015 Pamplona, Spain. Email address: [email protected]
∗∗∗ Postal address: Departamento de Métodos Estadísticos, Facultad de Matemáticas, Pedro Cerbuna, 12, 50009 Zaragoza, Spain.
∗∗∗ Postal address: Departamento de Métodos Estadísticos, Facultad de Matemáticas, Pedro Cerbuna, 12, 50009 Zaragoza, Spain.
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Abstract

The aim of this paper is to study the distribution of colours, {Xn}, in a generalized Pólya urn model with L colours, an urn function and a random environment. In this setting, the number of actions to be taken can be greater than L, and the total number of balls added in each step can be random. The process {Xn} is expressed as a stochastic recurrent equation that fits a Robbins—Monro scheme. Since this process evolves in the (L—1)-simplex, the stability of the solutions of the ordinary differential equation associated with the Robbins—Monro scheme can be studied by means of differential algebraic equations. This approach provides a method of obtaining strong laws for the process {Xn}.

Keywords

Urn modelsRobbins—Monro algorithmODE methoddifferential algebraic equations

MSC classification

Primary: 60F15: Strong theorems
Secondary: 34D05: Asymptotic properties
Type
Research Papers
Information
Journal of Applied Probability , Volume 40 , Issue 2 , June 2003 , pp. 401 - 412
Copyright
Copyright © Applied Probability Trust 2003 

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Footnotes

Partly supported by P071/2000 project of D.G.A. and BFM2001-2449 project of CICYT.

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