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A TIME-DEPENDENT PÓLYA URN WITH MULTIPLE DRAWINGS

Published online by Cambridge University Press:  27 March 2019

May-Ru Chen
May-Ru Chen*
Affiliation:
Department of Applied Mathematics, National Sun Yat-sen University, 70 Lien-hai Rd., Kaohsiung 804, Taiwan, ROC E-mail: mayru@faculty.nsysu.edu.tw
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Abstract

In this paper, we consider a generalized Pólya urn model with multiple drawings and time-dependent reinforcements. Suppose an urn initially contains w white and r red balls. At the nth action, m balls are drawn at random from the urn, say k white and m−k red balls, and then replaced in the urn along with cnk white and cn(m − k) red balls, where {cn} is a given sequence of positive integers. Repeat the above procedure ad infinitum. Let Xn be the proportion of the white balls in the urn after the nth action. We first show that Xn converges almost surely to a random variable X. Next, we give a necessary and sufficient condition for X to have a Bernoulli distribution with parameter w/(w + r). Finally, we prove that X is absolutely continuous if {cn} is bounded.

Keywords

absolutely continuousatomsmartingaleurn model
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 34 , Issue 4 , October 2020 , pp. 469 - 483
Copyright
Copyright © Cambridge University Press 2019

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