Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-08T21:37:30.677Z Has data issue: false hasContentIssue false
  • English
  • Français

A new urn model

Part of: Limit theorems Stochastic processes Distribution theory - Probability

Published online by Cambridge University Press:  14 July 2016

May-Ru Chen  and
Ching-Zong Wei
May-Ru Chen*
Affiliation:
National Changhua University of Education
Ching-Zong Wei*
Affiliation:
Academia Sinica, Taiwan
*
∗Postal address: Department of Mathematics, National Changhua University of Education, 1 Jin-De Road, Changhua, 500, Taiwan, Republic of China. Email address: [email protected]
∗∗Postal address: Institute of Statistical Science, Academia Sinica, 128 Academia Road, Sec. 2, Taipei, 115, Taiwan, Republic of China. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we propose a new urn model. A single urn contains b black balls and w white balls. For each observation, we randomly draw m balls and note their colors, say k black balls and m − k white balls. We return the drawn balls to the urn with an additional ck black balls and c(m − k) white balls. We repeat this procedure n times and denote by Xn the fraction of black balls after the nth draw. To investigate the asymptotic properties of Xn, we first perform some computational studies. We then show that {Xn} forms a martingale, which converges almost surely to a random variable X. The distribution of X is then shown to be absolutely continuous.

Keywords

Urn modelmartingalesymmetric polynomialabsolute continuity for a random variable

MSC classification

Primary: 60E05: Distributions
Secondary: 60G42: Martingales with discrete parameter 60F15: Strong theorems
Type
Research Papers
Information
Journal of Applied Probability , Volume 42 , Issue 4 , December 2005 , pp. 964 - 976
Copyright
© Applied Probability Trust 2005 

References

Bagchi, A. and Pal, A. K. (1985). Asymptoic normality in the generalized Pólya–Eggenberger urn model, with an application to computer data structures. SIAM J. Algebraic Discrete Meth. 6, 394–405.CrossRefGoogle Scholar
Billingsley, P. (1995). Probability and Measure, 3rd edn. John Wiley, New York.Google Scholar
Eggenberger, F. and Pólya, G. (1923). Über die statistik verketteter vorgänge. Z. Angewandte Math. Mech. 1, 279–289.Google Scholar
Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edn. John Wiley, New York.Google Scholar
Gouet, R. (1989). A martingale approach to strong convergence in a generalized Pólya–Eggenberger urn model. Statist. Prob. Lett. 8, 225–228.Google Scholar
Gouet, R. (1993). Martingale functional central limit theorems for a generalized Pólya urn. Ann. Prob. 21, 1624–1639.CrossRefGoogle Scholar
Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and Its Application. Academic Press, New York.Google Scholar
Hill, B., 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 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.Google Scholar
Maistrov, L. E. (1974). Probability Theory: a Historical Sketch. Academic Press, New York.Google Scholar
Pemantle, R. (1990). A time-dependent version of Pólya's urn. J. Theoret. Prob. 3, 627–637.Google Scholar
Schreiber, S. J. (2001). Urn models, replicator processes, and random genetic drift. SIAM J. Appl. Math. 61, 2148–2167.Google Scholar
Stewart, I. (1989). Galois Theory, 2nd edn. Chapman and Hall, New York.Google Scholar
Wei, C. Z. (1993). Martingale transforms with non-atomic limits and stochastic approximation. Prob. Theory Relat. Fields 95, 103–114.CrossRefGoogle Scholar
Wheeden, R. L. and Zygmund, A. (1977). Measure and Integral. Marcel Dekker, New York.Google Scholar