The simple harmonic urn
(2011) In Annals of Probability 39(6). p.21192177 Abstract
 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... (More)
 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 continuoustime 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 twodimensional 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. (Less)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/record/2296053
 author
 Crane, Edward; Georgiou, Nicolas; Volkov, Stanislav ^{LU} ; Wade, Andrew R. and Waters, Robert J.
 organization
 publishing date
 2011
 type
 Contribution to journal
 publication status
 published
 subject
 keywords
 Urn model, recurrence classification, oriented percolation, uniform renewal process, twodimensional linear birth and death process, Bessel process, coupling, Eulerian numbers
 in
 Annals of Probability
 volume
 39
 issue
 6
 pages
 2119  2177
 publisher
 Institute of Mathematical Statistics
 external identifiers

 scopus:84865092507
 ISSN
 00911798
 DOI
 10.1214/10AOP605
 language
 English
 LU publication?
 yes
 id
 32c72918338c4161813016edb13275bd (old id 2296053)
 date added to LUP
 20120120 16:03:23
 date last changed
 20170611 04:20:06
@article{32c72918338c4161813016edb13275bd, abstract = {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 continuoustime 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 twodimensional 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.}, author = {Crane, Edward and Georgiou, Nicolas and Volkov, Stanislav and Wade, Andrew R. and Waters, Robert J.}, issn = {00911798}, keyword = {Urn model,recurrence classification,oriented percolation,uniform renewal process,twodimensional linear birth and death process,Bessel process,coupling,Eulerian numbers}, language = {eng}, number = {6}, pages = {21192177}, publisher = {Institute of Mathematical Statistics}, series = {Annals of Probability}, title = {The simple harmonic urn}, url = {http://dx.doi.org/10.1214/10AOP605}, volume = {39}, year = {2011}, }