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The simple harmonic urn

Crane, Edward; Georgiou, Nicolas; Volkov, Stanislav LU ; Wade, Andrew R. and Waters, Robert J. (2011) In Annals of Probability 39(6). p.2119-2177
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 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. (Less)
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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Urn model, recurrence classification, oriented percolation, uniform renewal process, two-dimensional 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
0091-1798
DOI
10.1214/10-AOP605
language
English
LU publication?
yes
id
32c72918-338c-4161-8130-16edb13275bd (old id 2296053)
date added to LUP
2012-01-20 16:03:23
date last changed
2017-06-11 04:20:06
@article{32c72918-338c-4161-8130-16edb13275bd,
  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 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.},
  author       = {Crane, Edward and Georgiou, Nicolas and Volkov, Stanislav and Wade, Andrew R. and Waters, Robert J.},
  issn         = {0091-1798},
  keyword      = {Urn model,recurrence classification,oriented percolation,uniform renewal process,two-dimensional linear birth and death process,Bessel process,coupling,Eulerian numbers},
  language     = {eng},
  number       = {6},
  pages        = {2119--2177},
  publisher    = {Institute of Mathematical Statistics},
  series       = {Annals of Probability},
  title        = {The simple harmonic urn},
  url          = {http://dx.doi.org/10.1214/10-AOP605},
  volume       = {39},
  year         = {2011},
}