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A new urn model

Part of: Stochastic processes Distribution theory - Probability Limit theorems

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: mayru@ms29.url.com.tw
∗∗Postal address: Institute of Statistical Science, Academia Sinica, 128 Academia Road, Sec. 2, Taipei, 115, Taiwan, Republic of China. Email address: czw@stat.sinica.edu.tw
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Abstract

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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 

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