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A New Two-Urn Model

Part of: Distribution theory - Probability Stochastic processes Limit theorems

Published online by Cambridge University Press:  19 February 2016

May-Ru Chen ,
Shoou-Ren Hsiau  and
Ting-Hsin Yang
May-Ru Chen*
Affiliation:
National Sun Yat-sen University
Shoou-Ren Hsiau*
Affiliation:
National Sun Yat-sen University
Ting-Hsin Yang*
Affiliation:
National Changhua University of Education
*
∗ Postal address: Department of Applied Mathematics, National Sun Yat-sen University, 70 Lien-hai Rd., Kaohsiung 804, Taiwan, R. O. C. Email address: mayru@faculty.nsysu.edu.tw.
∗∗ Postal address: Department of Mathematics, National Changhua University of Education, No. 1 Jin-De Road, Changhua 500, Taiwan, R. O. C.
∗∗ Postal address: Department of Mathematics, National Changhua University of Education, No. 1 Jin-De Road, Changhua 500, Taiwan, R. O. C.
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Abstract

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We propose a two-urn model of Pólya type as follows. There are two urns, urn A and urn B. At the beginning, urn A contains rA red and wA white balls and urn B contains rB red and wB white balls. We first draw m balls from urn A and note their colors, say i red and m - i white balls. The balls are returned to urn A and bi red and b(m - i) white balls are added to urn B. Next, we draw ℓ balls from urn B and note their colors, say j red and ℓ - j white balls. The balls are returned to urn B and aj red and a(ℓ - j) white balls are added to urn A. Repeat the above action n times and let Xn be the fraction of red balls in urn A and Yn the fraction of red balls in urn B. We first show that the expectations of Xn and Yn have the same limit, and then use martingale theory to show that Xn and Yn converge almost surely to the same limit.

Keywords

Two-urn modelmartingale

MSC classification

Primary: 60F15: Strong theorems
Secondary: 60G42: Martingales with discrete parameter 60E05: Distributions
Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 2 , June 2014 , pp. 590 - 597
Copyright
© Applied Probability Trust 

References

Athreya, K. B. and Lahiri, S. N. (2006). Measure Theory and Probability Theory. Springer, New York.Google 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. Algebraic Discrete Methods 6, 394–405.Google Scholar
Bernard, S. R., Sobel, M. and Uppuluri, V. R. R. (1981). On a two urn model of Pólya-type. Bull. Math. Biol. 43, 33–45.Google Scholar
Chen, M.-R. and Wei, C.-Z. (2005). A new urn model. J. Appl. Prob. 42, 964–976.CrossRefGoogle Scholar
Eggenberger, F. and Pólya, G. (1923). Über die Statistik verketetter Vorgänge. Zeitschrift für Angewandte Mathematik und Mechanik. 3, 279–289.Google Scholar
Gouet, R. (1989). A martingale approach to strong convergence in a generalized Pólya–Eggenberger urn model. Statis. Prob. Lett. 8, 225–228.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.Google 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
Johnson, N. L. and Kotz, S. (1977). Urn Models and Their Application, 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, ed. Balakrishnan, N., Birkhäuser, Boston, pp. 203–257.Google Scholar
Mahmoud, H. M. (2009). Pólya Urn Models, CRC Press, Boca Raton, FL.Google Scholar
Martin, C. F. and Allen, L. J. S. (1995). Urn model simulations of a sexually transmitted disease epidemic. Appl. Math. Comput. 71, 179–199.Google Scholar
Pemantle, R. (1990). A time-dependent version of Pólya's urn. J. Theoret. Prob. 3, 627–637.Google Scholar
Renlund, H. (2010). Generalized Pólya urns via stochastic approximation. Preprint, available at {http://arxiv.org/abs/1002.3716}.Google Scholar