Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-24T17:44:47.423Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected]
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Aguech, R. (2009).Limit theorems for random triangular urn schemes.J. Appl. Prob. 46,827–843.Google Scholar
[2] Antal, T.,Ben-Naim, E. and Krapivsky, P. L. (2010).First-passage properties of the Pólya urn process.J. Statist. Mech. Theory Exp. 2010,PO7009.Google Scholar
[3] Athreya, K. B. and Ney, P. E. (1972).Branching Processes.Springer ,New York.Google Scholar
[4] Backhausz, Á. and Móri, T. F. (2015).Asymptotic properties of a random graph with duplications.J. Appl. Prob. 52,375–390.Google Scholar
[5] Backhausz, Á. and Móri, T. F.(2016).Further properties of a random graph with duplications and deletions.Stoch. Models 32,99–120.Google Scholar
[6] Champagnat, N.,Lambert, A. and Richard, M. (2012).Birth and death processes with neutral mutations.Internat. J. Stoch. Anal. 2012,569081.Google Scholar
[7] Chung, F.,Handjani, S. and Jungreis, D. (2003).Generalizations of Pólya’s urn problem.Ann. Combinatorics 7,141–153.Google Scholar
[8] Dereich, S. and Mörters, P. (2009).Random networks with sublinear preferential attachment: degree evolutions.Electron. J. Prob. 14,1222–1267.Google Scholar
[9] Galashin, P. (2014).Existence of a persistent hub in the convex preferential attachment model. Preprint. Available at http://arxiv.org/abs/1310.7513v3.Google Scholar
[10] Ivanova, A.,Rosenberger, W. F.,Durham, S. D. and Flournoy, F. (2000).A birth and death urn for randomized clinical trials: asymptotic methods.Sankhyā B 62,104–118.Google Scholar
[11] Janson, S. (2004).Functional limit theorems for multitype branching processes and generalized Pólya urns.Stoch. Process. Appl. 110,177–245.Google Scholar
[12] Janson, S. (2006).Limit theorems for triangular urn schemes.Prob. Theory Relat. Fields 134,417–452.Google Scholar
[13] Khanin, K. and Khanin, R. (2001).A probabilistic model for the establishment of neuron polarity.J. Math. Biol. 42,26–40.CrossRefGoogle ScholarPubMed
[14] Kuba, M. and Panholzer, A. (2012).Limiting distributions for a class of diminishing urn models.Adv. Appl. Prob. 44,87–116.Google Scholar
[15] Rösler, U. and Rüschendorf, L. (2001).The contraction method for recursive algorithms.Algorithmica 29,3–33.Google Scholar
[16] Rüschendorf, L. (2006).On stochastic recursive equations of sum and max type.J. Appl. Prob. 43,687–703.Google Scholar
[17] Thörnblad, E. (2015).Asymptotic degree distribution of a duplication-deletion random graph model.Internet Math. 11,289–305.CrossRefGoogle Scholar
[18] Wallstrom, T. C. (2012).The equalization probability of the Pólya urn.Amer. Math. Monthly 119,516–518.Google Scholar