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On Generalized Pólya Urn Models

Part of: Graph theory Limit theorems Combinatorics

Published online by Cambridge University Press:  30 January 2018

May-Ru Chen  and
Markus Kuba
May-Ru Chen*
Affiliation:
National Sun Yat-sen University
Markus Kuba*
Affiliation:
Technische Universität Wien
*
∗ Postal address: Department of Applied Math., National Sun Yat-sen University, 70 Lien-hai Road, Kaohsiung 804, Taiwan, R.O.C. Email address: mayru@faculty.nsysu.edu.tw
∗∗ Postal address: Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Wiedner Hauptstr. 8-10/104, 1040 Wien, Austria, HTL Wien 5 Spengergasse, Spengergasse 20, 1050 Wien, Austria. Email address: kuba@dmg.tuwien.ac.at
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Abstract

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We study an urn model introduced in the paper of Chen and Wei (2005), where at each discrete time step m balls are drawn at random from the urn containing colors white and black. Balls are added to the urn according to the inspected colors, generalizing the well known Pólya-Eggenberger urn model, case m = 1. We provide exact expressions for the expectation and the variance of the number of white balls after n draws, and determine the structure of higher moments. Moreover, we discuss extensions to more than two colors. Furthermore, we introduce and discuss a new urn model where the sampling of the m balls is carried out in a step-by-step fashion, and also introduce a generalized Friedman's urn model.

Keywords

Urn modellimiting distribution

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 05A15: Exact enumeration problems, generating functions 05C05: Trees

Information

Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 4 , December 2013 , pp. 1169 - 1186
Copyright
© Applied Probability Trust 

References

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. Algebraic Discrete Math. 6, 394–405.CrossRefGoogle Scholar
Chen, M.-R. and Wei, C.-Z. (2005). A new urn model. J. Appl. Prob. 42, 964–976.CrossRefGoogle Scholar
Chen, M.-R., Hsiau, S.-R. and Yang, T.-H. (2012). A new two-urn model. To appear in J. Appl. Prob. Google Scholar
Eggenberger, F. and Pólya, G. (1923). Über die Statistik verketteter Vorgänge. Z. Angewandte Math. Mech. 1, 279–289.Google Scholar
Flajolet, P., Dumas, P. and Puyhaubert, V. (2006). Some exactly solvable models of urn process theory. In Discrete Mathematics and Theoretical Computer Science (Proc. Fourth Colloquium Math. Comp. Sci.), ed. Chassaing, P., pp. 59–118.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
Graham, R. L., Knuth, D. E. and Patashnik, O. (1994). Concrete Mathematics. Addison-Wesley.Google Scholar
Hill, B., Lane, D. and Sudderth, W. (1980). A strong law for some generalized urn processes. Ann. Prob. 8, 214–226.Google Scholar
Janson, S. (2006). Limit theorems for triangular urn schemes. Prob. Theory Relat. Fields 134, 417–452.Google Scholar
Johnson, N. L. and Kotz, S. (1977). Urn Models and Their Application. John Wiley, New York.Google Scholar
Kotz, S. and Balakrishnan, N. (1997). Advances in urn models during the past two decades. In Advances in Combinatorial Methods and Applications to Probability and Statistics, Birkhäuser, Boston, MA, pp. 203–257.CrossRefGoogle Scholar
Mahmoud, H. (2009). Pólya Urn Models. CRC Press, Boca Raton.Google Scholar
Pemantle, R. (1990). A time-dependent version of Pólya's urn. J. Theoret. Prob. 3, 627–637.CrossRefGoogle Scholar
Renlund, H. (2010). Generalized Pólya urns via stochastic approximation. Availaible at http://arxiv.org/abs/1002.3716.Google Scholar
Schreiber, S. J. (2001). Urn models, replicator processes, and random genetic drift. SIAM J. Appl. Math. 61, 2148–2167.Google Scholar