Journal of Applied Probability

A new two-urn model

May-Ru Chen, Shoou-Ren Hsiau, and Ting-Hsin Yang

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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.

Article information

J. Appl. Probab., Volume 51, Number 2 (2014), 590-597.

First available in Project Euclid: 12 June 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F15: Strong theorems
Secondary: 60G42: Martingales with discrete parameter 60E05: Distributions: general theory

Two-urn model martingale


Chen, May-Ru; Hsiau, Shoou-Ren; Yang, Ting-Hsin. A new two-urn model. J. Appl. Probab. 51 (2014), no. 2, 590--597. doi:10.1239/jap/1402578645.

Export citation


  • Athreya, K. B. and Lahiri, S. N. (2006). Measure Theory and Probability Theory. Springer, New York.
  • 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.
  • 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.
  • Chen, M.-R. and Wei, C.-Z. (2005). A new urn model. J. Appl. Prob. 42, 964–976.
  • Eggenberger, F. and Pólya, G. (1923). Über die Statistik verketetter Vorgänge. Zeitschrift für Angewandte Mathematik und Mechanik. 3, 279–289.
  • Gouet, R. (1989). A martingale approach to strong convergence in a generalized Pólya–Eggenberger urn model. Statis. Prob. Lett. 8, 225–228.
  • Higueras, I., Moler, J., Plo, F. and San Miguel, M. (2003). Urn models and differential algebraic equations. J. Appl. Prob. 40, 401–412.
  • Hill, B. M., Lane, D. and Sudderth, W. (1980). A strong law for some generalized urn processes. Ann. Prob. 8, 214–226.
  • Johnson, N. L. and Kotz, S. (1977). Urn Models and Their Application, Wiley, New York.
  • 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. N. Balakrishnan, Birkhäuser, Boston, pp. 203–257.
  • Mahmoud, H. M. (2009). Pólya Urn Models, CRC Press, Boca Raton, FL.
  • Martin, C. F. and Allen, L. J. S. (1995). Urn model simulations of a sexually transmitted disease epidemic. Appl. Math. Comput. 71, 179–199.
  • Pemantle, R. (1990). A time-dependent version of Pólya's urn. J. Theoret. Prob. 3, 627–637.
  • Renlund, H. (2010). Generalized Pólya urns via stochastic approximation. Preprint, available at 1002.3716.