Open Access
November 2011 The simple harmonic urn
Edward Crane, Nicholas Georgiou, Stanislav Volkov, Andrew R. Wade, Robert J. Waters
Ann. Probab. 39(6): 2119-2177 (November 2011). DOI: 10.1214/10-AOP605


We study a generalized Pólya urn model with two types of ball. If the drawn ball is red, it is replaced together with a black ball, but if the drawn ball is black it is replaced and a red ball is thrown out of the urn. When only black balls remain, the roles of the colors are swapped and the process restarts. We prove that the resulting Markov chain is transient but that if we throw out a ball every time the colors swap, the process is recurrent. We show that the embedded process obtained by observing the number of balls in the urn at the swapping times has a scaling limit that is essentially the square of a Bessel diffusion. We consider an oriented percolation model naturally associated with the urn process, and obtain detailed information about its structure, showing that the open subgraph is an infinite tree with a single end. We also study a natural continuous-time embedding of the urn process that demonstrates the relation to the simple harmonic oscillator; in this setting, our transience result addresses an open problem in the recurrence theory of two-dimensional linear birth and death processes due to Kesten and Hutton. We obtain results on the area swept out by the process. We make use of connections between the urn process and birth–death processes, a uniform renewal process, the Eulerian numbers, and Lamperti’s problem on processes with asymptotically small drifts; we prove some new results on some of these classical objects that may be of independent interest. For instance, we give sharp new asymptotics for the first two moments of the counting function of the uniform renewal process. Finally, we discuss some related models of independent interest, including a “Poisson earthquakes” Markov chain on the homeomorphisms of the plane.


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Edward Crane. Nicholas Georgiou. Stanislav Volkov. Andrew R. Wade. Robert J. Waters. "The simple harmonic urn." Ann. Probab. 39 (6) 2119 - 2177, November 2011.


Published: November 2011
First available in Project Euclid: 17 November 2011

zbMATH: 1262.60070
MathSciNet: MR2932666
Digital Object Identifier: 10.1214/10-AOP605

Primary: 60J10
Secondary: 60J25 , 60K05 , 60K35

Keywords: Bessel process , coupling , Eulerian numbers , Oriented percolation , recurrence classification , two-dimensional linear birth and death process , uniform renewal process , urn model

Rights: Copyright © 2011 Institute of Mathematical Statistics

Vol.39 • No. 6 • November 2011
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