Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-27T06:28:55.475Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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 

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.)

Footnotes

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

References

Artur, B., Ermoléev, Y., and Kaniovskii, Y. (1988). 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
Bagchi, A., and Pal, A. K. (1985). Asymptotic normality in the generalized Pólya-Eggenberger urn model, with an application to computer data structures}. SIAM J. Alg. Discrete Meth. 6, 394–405.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
Duflo, M. (1997). Random Iterative Models. Springer, Berlin.Google Scholar
Eggenberger, F. and Pólya, G. (1928). Sur l'interprétation de certaines courbes de fréquence. C. R. Acad. Sci. Paris 187, 870–872.Google Scholar
Freedman, D. (1965). Bernard Friedman's urn}. Ann. Math. Statist. 36, 956–970.CrossRefGoogle Scholar
Friedman, B. (1949). A simple urn model. Commun. Pure Appl. Math. 2, 59–70.CrossRefGoogle Scholar
Gouet, R. (1989). A martingale approach to strong convergence in a generalized Pólya-Eggenberger urn model}. Statist. Prob. Lett. 8, 225–228.CrossRefGoogle Scholar
Gouet, R. (1993). Martingale functional central limit theorems for a generalized Pólya urn}. Ann. Prob. 21, 1624–1639.CrossRefGoogle Scholar
Gouet, R. (1997). Strong convergence of proportions in a multicolor Pólya urn}. J. Appl. Prob. 34, 426–435.CrossRefGoogle Scholar
Griepentrog, E. and März, R. (1986). Differential Algebraic Equations and Their Numerical Treatment. Teubner, Leipzig.Google Scholar
Hall, P., and Heyde, C. C. (1980). Martingale Limit Theory and Its Applications. Academic Press, San Diego.Google Scholar
Higueras, I. and García-Celayeta, B. (1997). Stability for linear constant DAEs. Tech. Rep., Departamento de Matemática e Informática, Universidad Pública de Navarra.Google Scholar
Hill, B. M., Lane, D., and Sudderth, W. (1980). A strong law for some generalized urn processes. Ann. Prob. 8, 214–226.CrossRefGoogle Scholar
Johnson, N. L., and Kotz, S. (1977). Urn Models and Their Applications. John Wiley, New York.Google Scholar
Kotz, S., Mahmoud, H., and Robert, P. (2000). On generalized Pólya urn models. Statist. Prob. Lett. 49, 163–173.CrossRefGoogle Scholar
Kushner, H. J., and Yin, G. G. (1997). Stochastic Approximation Algorithms and Applications. Springer, New York.CrossRefGoogle Scholar
März, R. (1992). On quasilinear index 2 differential-algebraic equations. In Berlin Seminar on Differential-Algebraic Equations (Seminarberichte 92-1), eds Griepentrog, E., Hanke, M. and März, R., Humboldt Universität Fachbereich Mathematik, Berlin, pp. 39–60. Available at http://www.mathematik.hu-berlin.de/publ/.Google Scholar
Pemantle, R. (1988). Phase transition in reinforced random walk and RWRE on trees}. Ann. Prob. 16, 1229–1241.CrossRefGoogle Scholar
Pemantle, R. (1990). Nonconvergence to unstable points in urn models and stochastic approximations}. Ann. Prob. 18, 698–712.CrossRefGoogle Scholar
Reich, S. (1995). On the local qualitative behavior of differential algebraic equations. Circuits Systems Signal Process. 14, 427–443.CrossRefGoogle Scholar
Schreiber, S. J. (2001). Urn models, replicator processes, and random genetic drift. SIAM J. Appl. Math. 61, 2148–2167.CrossRefGoogle Scholar
Seneta, E. (1981). Nonnegative Matrices and Markov Chains. Springer, New York.CrossRefGoogle Scholar
Smythe, R. T. (1996). Central limit theorems for urn models. Stoch. Process. Appl. 65, 115–137.CrossRefGoogle Scholar
Stuart, A. M., and Humphries, A. R. (1996). Dynamical Systems and Numerical Analysis. Cambridge University Press. New York.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.CrossRefGoogle Scholar