Home

# The dominating colour of an infinite Pólya urn model

## 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).

## Corresponding author

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

## References

Hide All
[1] Aguech, R. (2009).Limit theorems for random triangular urn schemes.J. Appl. Prob. 46,827843.
[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.
[3] Athreya, K. B. and Ney, P. E. (1972).Branching Processes.Springer ,New York.
[4] Backhausz, Á. and Móri, T. F. (2015).Asymptotic properties of a random graph with duplications.J. Appl. Prob. 52,375390.
[5] Backhausz, Á. and Móri, T. F.(2016).Further properties of a random graph with duplications and deletions.Stoch. Models 32,99120.
[6] Champagnat, N.,Lambert, A. and Richard, M. (2012).Birth and death processes with neutral mutations.Internat. J. Stoch. Anal. 2012,569081.
[7] Chung, F.,Handjani, S. and Jungreis, D. (2003).Generalizations of Pólya’s urn problem.Ann. Combinatorics 7,141153.
[8] Dereich, S. and Mörters, P. (2009).Random networks with sublinear preferential attachment: degree evolutions.Electron. J. Prob. 14,12221267.
[9] Galashin, P. (2014).Existence of a persistent hub in the convex preferential attachment model. Preprint. Available at http://arxiv.org/abs/1310.7513v3.
[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 62,104118.
[11] Janson, S. (2004).Functional limit theorems for multitype branching processes and generalized Pólya urns.Stoch. Process. Appl. 110,177245.
[12] Janson, S. (2006).Limit theorems for triangular urn schemes.Prob. Theory Relat. Fields 134,417452.
[13] Khanin, K. and Khanin, R. (2001).A probabilistic model for the establishment of neuron polarity.J. Math. Biol. 42,2640.
[14] Kuba, M. and Panholzer, A. (2012).Limiting distributions for a class of diminishing urn models.Adv. Appl. Prob. 44,87116.
[15] Rösler, U. and Rüschendorf, L. (2001).The contraction method for recursive algorithms.Algorithmica 29,333.
[16] Rüschendorf, L. (2006).On stochastic recursive equations of sum and max type.J. Appl. Prob. 43,687703.
[17] Thörnblad, E. (2015).Asymptotic degree distribution of a duplication-deletion random graph model.Internet Math. 11,289305.
[18] Wallstrom, T. C. (2012).The equalization probability of the Pólya urn.Amer. Math. Monthly 119,516518.

# The dominating colour of an infinite Pólya urn model

## Metrics

### Full text viewsFull text views reflects the number of PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 0 *