The dominating colour of an infinite Pólya urn model
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Publication:2836240
DOI10.1017/JPR.2016.49zbMATH Open1351.60056arXiv1506.05862OpenAlexW3099384793MaRDI QIDQ2836240
Publication date: 9 December 2016
Published in: Journal of Applied Probability (Search for Journal in Brave)
Abstract: We study a P'olya-type urn model defined as follows. Start at time 0 with a single ball of some colour. Then, at each time n>0, choose a ball from the urn uniformly at random. With probability 1/2<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 M'ori, and Th"ornblad. 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 P'olya urn model (which corresponds to p=1).
Full work available at URL: https://arxiv.org/abs/1506.05862
Sums of independent random variables; random walks (60G50) Branching processes (Galton-Watson, birth-and-death, etc.) (60J80)
Related Items (2)
A Pólya urn model with a continuum of colors ⋮ Strong convergence of infinite color balanced urns under uniform ergodicity
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