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The dominating colour of an infinite Pólya urn model

Part of: Stochastic processes Markov processes

Published online by Cambridge University Press:  24 October 2016

Erik Thörnblad
Erik Thörnblad*
Affiliation:
Uppsala University
*
* Postal address: Department of Mathematics, Uppsala University, Box 480, S-75106 Uppsala, Sweden. Email address: erik.thornblad@math.uu.se
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Abstract

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).

Keywords

Urn modellargest colourrandom graphpersistent hub

MSC classification

Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 53 , Issue 3 , September 2016 , pp. 914 - 924
Copyright
Copyright © Applied Probability Trust 2016 

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