The simple harmonic urn

@article{Crane2011TheSH,
  title={The simple harmonic urn},
  author={Edward Crane and Nicholas Georgiou and Stanislav Volkov and Andrew R. Wade and Robert J. Waters},
  journal={Annals of Probability},
  year={2011},
  volume={39},
  pages={2119-2177}
}
We study a generalized Polya 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… Expand

Figures from this paper

Pólya Urn Schemes with Infinitely Many Colors
In this work we introduce a new urn model with infinite but countably many colors indexed by an appropriate infinite set. We mainly focus on d-dimensional integer lattice and replacement matrixExpand
Linear de-preferential urn models
Abstract In this paper we consider a new type of urn scheme, where the selection probabilities are proportional to a weight function, which is linear but decreasing in the proportion of existingExpand
Limiting Distributions for a Class Of Diminishing Urn Models
In this work we analyze a class of 2 × 2 Pólya-Eggenberger urn models with ball replacement matrix and c = pa with . We determine limiting distributions by obtaining a precise recursive descriptionExpand
Excursions and path functionals for stochastic processes with asymptotically zero drifts
We study discrete-time stochastic processes (Xt) on [0,∞) with asymptotically zero mean drifts. Specifically, we consider the critical (Lamperti-type) situation in which the mean drift at x is aboutExpand
Generalized P\'olya Urn Schemes with Negative but Linear Reinforcements
In this paper, we consider a new type of urn scheme, where the selection probabilities are proportional to a weight function, which is linear but decreasing in the proportion of existing colours. WeExpand
A New Approach to Pólya Urn Schemes and Its Infinite Color Generalization
In this work we generalize Polya urn schemes with possibly infinitely many colors and extend the earlier models described in [4, 5, 7]. We provide a novel and unique approach of representing theExpand
Linear competition processes and generalized Pólya urns with removals
A competition process is a continuous time Markov chain that can be interpreted as a system of interacting birth-and-death processes, the components of which evolve subject to a competitiveExpand
Impatient Random Walk
We introduce a new type of random walk where the definition of edge repellence/reinforcement is very different from the one in the “traditional” reinforced random walk models and investigate itsExpand
Waiting Times for Ties in Random Competitions
ABSTRACT. Multiple teams participate in a random competition. In each round the winner receives one point. We study the times until ties occur among teams. The martingales and supermartingales thatExpand
...
1
2
...

References

SHOWING 1-10 OF 49 REFERENCES
AN URN MODEL FOR CANNIBAL BEHAVIOR
A sampling procedure involving an urn with red and white balls in it is studied. Initially, the urn contains n balls, r of them being white. At each step, a white ball is removed, and one more ballExpand
Critical random walks on two-dimensional complexes with applications to polling systems
We consider a time-homogeneous random walk Xi = {xi (t)} on a two-dimensional complex. All of our results here are formulated in a constructive way. By this we mean that for any given random walk weExpand
Vertex-reinforced random walk on Z has finite range
A stochastic process called vertex-reinforced random walk (VRRW) is defined in Pemantle [Ann. Probab. 16 1229-1241]. We consider this process in the case where the underlying graph is an infiniteExpand
Polya Urn Models
Incorporating a collection of recent results, Plya Urn Models deals with discrete probability through the modern and evolving urn theory and its numerous applications. The book first substantiatesExpand
Functional limit theorems for multitype branching processes and generalized Pólya urns
A functional limit theorem is proved for multitype continuous time Markov branching processes. As consequences, we obtain limit theorems for the branching process stopped by some stopping rule, forExpand
Martingales in the Ok Corral
In the model of the OK Corral formulated by Williams and McIlroy [ 2 ]: ‘Two lines of gunmen face each other, there being initially m on one side, n on the other. Each person involved is a hopelessExpand
Tails of passage-times and an application to stochastic processes with boundary reflection in wedges
In this paper we obtain lower bounds for the tails of the distributions of the first passage-times for some stochastic processes. We consider first discrete parameter processes with asymptoticallyExpand
Asymptotic Behaviour of Randomly Reflecting Billiards in Unbounded Tubular Domains
We study stochastic billiards in infinite planar domains with curvilinear boundaries: that is, piecewise deterministic motion with randomness introduced via random reflections at the domain boundary.Expand
THE RECURRENCE AND TRANSIENCE OF TWO-DIMENSIONAL LINEAR BIRTH AND DEATH PROCESSES
A two-dimensional linear birth and death process is a continuous-time Markov chain Y(-) with state space (Z,)2 which can jump from the point (n, m) to one of its four neighbors, with rates that areExpand
Convergence of independent particle systems
Abstract We consider a system of particles moving independently on a countable state space, according to a general (non-space-homogeneous) Markov process. Under mild conditions, the number ofExpand
...
1
2
3
4
5
...