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September 2016 The dominating colour of an infinite Pólya urn model
Erik Thörnblad
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J. Appl. Probab. 53(3): 914-924 (September 2016).


We study a Pólya-type urn model defined as follows. Start at time 0 with a single ball of some colour. Then, at each time n≥1, choose a ball from the urn uniformly at random. With probability ½<p<1, return the ball to the urn along with another ball of the same colour. With probability 1−p, recolour the ball to a new colour and then return it to the urn. This is equivalent to the supercritical case of a random graph model studied by Backhausz and Móri (2015), (2016) and Thörnblad (2015). We prove that, with probability 1, there is a dominating colour, in the sense that, after some random but finite time, there is a colour that always has the most number of balls. A crucial part of the proof is the analysis of an urn model with two colours, in which the observed ball is returned to the urn along with another ball of the same colour with probability p, and removed with probability 1−p. Our results here generalise a classical result about the Pólya urn model (which corresponds to p=1).


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Erik Thörnblad. "The dominating colour of an infinite Pólya urn model." J. Appl. Probab. 53 (3) 914 - 924, September 2016.


Published: September 2016
First available in Project Euclid: 13 October 2016

zbMATH: 1351.60056
MathSciNet: MR3570103

Primary: 60G50
Secondary: 60J80

Keywords: largest colour , persistent hub , random graph , urn model

Rights: Copyright © 2016 Applied Probability Trust


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Vol.53 • No. 3 • September 2016
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