Solution to the OK Corral Model via Decoupling of Friedman's Urn

@article{Kingman2003SolutionTT,
  title={Solution to the OK Corral Model via Decoupling of Friedman's Urn},
  author={J. F. C. Kingman and Stanislav Volkov},
  journal={Journal of Theoretical Probability},
  year={2003},
  volume={16},
  pages={267-276}
}
We consider the OK Corral model formulated by Williams and McIlroy(11) and later studied by Kingman.(7) In this paper we refine some of Kingman's results, by showing the connection between this model and Friedman's urn, and using Rubin's construction to decouple the urn. Also we obtain the exact expression for the probability of survival of exactly S gunmen given an initially fair configuration. 
Some exactly solvable models of urn process theory
We establish a fundamental isomorphism between discrete-time balanced urn processes and certain ordinary differential systems, which are nonlinear, autonomous, and of a simple monomial form. As a
On the Estimation of Parameter of Weighted Sums of Exponential Distribution
The random variable , with and   being independent exponentially distributed random variables with mean one, is considered. Van Leeuwaarden and Temme (2011) attempted to determine good approximation
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 description
On death processes and urn models
We use death processes and embeddings into continuous time in order to analyze several urn models with a diminishing content. In particular we discuss generalizations of the pill's problem,
On Sampling without replacement and OK-Corral urn models
In this work we discuss two urn models with general weight sequences $(A,B)$ associated to them, $A=(\alpha_n)_{n\in\N}$ and $B=(\beta_m)_{m\in\N}$, generalizing two well known P\'olya-Eggenberger
Border aggregation model
Start with a graph with a subset of vertices called {\it the border}. A particle released from the origin performs a random walk on the graph until it comes to the immediate neighbourhood of the
Boundary effect in competition processes
TLDR
It is proved that, with probability one, eventually one of the components of the Markov chain tends to infinity, while the remaining one oscillates between values $0$ and $1$.
Linear competition processes and generalized Pólya urns with removals
DAVID STENLUND: Hitting times in urn models and occupation times in one-dimensional diffusion models
The main subject of this thesis is certain functionals of Markov processes. The thesis can be said to consist of three parts. The first part concerns hitting times in urn models, which are Markov
...
1
2
3
4
...

References

SHOWING 1-10 OF 17 REFERENCES
Attracting edge property for a class of reinforced random walks
Using martingale techniques and comparison with the generalized Urn scheme, it is shown that the edge reinforced random walk on a graph of bounded degree, with the weight function W(k)=kρ,ρ>1,
The Ok Corral and the Power of the Law (A Curious Poisson‐Kernel Formula for a Parabolic Equation)
Two lines of gunmen face each other, there being initially m on one side, n on the other. Each person involved is a hopeless shot, but keeps firing at the enemy until either he himself is killed or
Probability and Measure. (2nd. ed
  • 1985
Sums of Independent Random Variables
I. Probability Distributions and Characteristic Functions.- 1. Random variables and probability distributions.- 2. Characteristic functions.- 3. Inversion formulae.- 4. The convergence of sequences
Sums of independent random variables. (2nd
  • 1975
Sums of independent random variables. (2nd. ed
  • 1975
A simple urn model
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 hopeless
Reinforced Random Walk
This thesis aim is to present results on a stochastic model called reinforced random walk. This process was conceived in the late 1980’s by Coppersmith and Diaconis and can be regarded as a
...
1
2
...