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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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References

1Billingsley, P. (1995). Probability and measure. 3rd ed. New York: John Wiley and Sons.Google Scholar
2Chen, M.-R. & Kuba, M. (2013). On generalized Pólya urn models. Journal of Applied Probability, 50: 1169–1186.10.1239/jap/1389370106CrossRefGoogle Scholar
3Chen, M.-R. & Wei, C.-Z. (2005). A new urn model. Journal of Applied Probability, 42: 964–976.10.1239/jap/1134587809CrossRefGoogle Scholar
4Eggenberger, F. & Pólya, G. (1923). Über die Statistik verketteter Vorgänge. Z. Angewandte Mathematics and Mechanics, 1: 279–289.10.1002/zamm.19230030407CrossRefGoogle Scholar
5Hall, P. & Heyde, C.C. (1980). Martingale limit theory and its application. New York: Academic Press.Google Scholar
6Hayhoe, M., Alajaji, F., & Gharesifard, B. (2018). A polya contagion model for networks. IEEE Transactions on Control of Network Systems, 5: 1998–2010.10.1109/TCNS.2017.2781467CrossRefGoogle Scholar
7Johnson, N.L. & Kotz, S (1977). Urn models and their application. New York: John Wiley and Sons, 176–181.Google Scholar
8Johnson, N., Kotz, S., & Mahmoud, H. (2004). Pólya-type urn models with multiple drawings. Journal of the Iranian Statistical Society, 3, 165–173.Google Scholar
9Kotz, S., & Balakrishnan, N. (1997). Advances in urn models during the past two decades. In Balakrishnan, N. (Ed.) Advances in combinatorial methods and applications to probability and statistics. Boston, MA: Birkhäuser, 203–257.10.1007/978-1-4612-4140-9_14CrossRefGoogle Scholar
10Kuba, M. & Mahmoud, H. (2015) Two-colour balanced affine urn models with multiple drawings I: central limit theorems, arXiv:1503.09069.Google Scholar
11Kuba, M., Mahmoud, H., & Panholzer, A. (2013). Analysis of a generalized Friedman's urn with multiple drawings. Discrete Applied Mathematics, 161: 2968–2984.10.1016/j.dam.2013.06.022CrossRefGoogle Scholar
12Mahmoud, H. (2009). Pólya urn models. Boca Raton, FL: CRC Press.Google Scholar
13Mahmoud, H. (2013). Drawing multisets of balls from tenable balanced linear urns. Probability in the Engineering and Informational Sciences, 27: 147–162.10.1017/S0269964812000381CrossRefGoogle Scholar
14Pemantle, R. (1990). A time-dependent version of Pólya's urn. Journal of Theoretical Probability, 3: 627–637.10.1007/BF01046101CrossRefGoogle Scholar
15Pemantle, R. (2007). A survey of random processes with reinforcement. Probability Surveys, 4: 1–79.10.1214/07-PS094CrossRefGoogle Scholar
16Rafik, A., Nabil, L., & Olfa, S. (2019). A generalized urn with multiple drawing and random addition. Annals of the Institute of Statistical Mathematics, 71: 389–408.10.1007/s10463-018-0651-3CrossRefGoogle Scholar
17Nabil, L., Mailler, C., & Olfa, S. (2018). Multiple drawing multi-colour urns by stochastic approximation. Journal of Applied Probability, 55: 254–281.Google Scholar