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A New Two-Urn Model

Part of: Distribution theory - Probability Stochastic processes Limit theorems

Published online by Cambridge University Press:  19 February 2016

May-Ru Chen ,
Shoou-Ren Hsiau  and
Ting-Hsin Yang
May-Ru Chen*
Affiliation:
National Sun Yat-sen University
Shoou-Ren Hsiau*
Affiliation:
National Sun Yat-sen University
Ting-Hsin Yang*
Affiliation:
National Changhua University of Education
*
∗ Postal address: Department of Applied Mathematics, National Sun Yat-sen University, 70 Lien-hai Rd., Kaohsiung 804, Taiwan, R. O. C. Email address: mayru@faculty.nsysu.edu.tw.
∗∗ Postal address: Department of Mathematics, National Changhua University of Education, No. 1 Jin-De Road, Changhua 500, Taiwan, R. O. C.
∗∗ Postal address: Department of Mathematics, National Changhua University of Education, No. 1 Jin-De Road, Changhua 500, Taiwan, R. O. C.
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Abstract

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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 r A red and w A white balls and urn B contains r B red and w B white balls. We first draw m balls from urn A and note their colors, say i red and m - iwhite 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 X n be the fraction of red balls in urn A and Y n the fraction of red balls in urn B. We first show that the expectations of X n and Y n have the same limit, and then use martingale theory to show that X n and Y n converge almost surely to the same limit.

Keywords

Two-urn modelmartingale

MSC classification

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

Information

Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 2 , June 2014 , pp. 590 - 597
Copyright
© Applied Probability Trust 

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